Jawaban quiz bagi kelas S1TI-1M
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Filed under: Linear Algebra and Matrices |
Reconfigurable Computing, Digital Signal Processing, Embedded Systems, System on Chip, and Algorithms 
<?
/*
nama : lutfi hanif
nim : 09.11.3507
kelas : SI TI 3M
*/
— mencari shuffle & contatination —
echo "”;
echo “”;
echo “”;
echo “lutfi hanif – 09.11.3507 – S1 TI 3M”;
echo “”;
echo “”;
echo “”;
echo ““;
echo “Lutfi hanif”;
echo “”;
echo “09.11.3507”;
echo “”;
echo “S1 TI 3M”;
echo “”;
echo ““;
echo “#########################”;
echo “”;
$proses = $_GET[‘proses’];
$proses = $_POST[‘submit’];
if(!isset($proses))
{
echo “”;
echo “”;
echo “”;
echo “”;
echo “masukkan kata pertama”;
echo “”;
echo “”;
echo “”;
echo “”;
echo “”;
echo “”;
echo “”;
echo “masukkan kata Kedua”;
echo “”;
echo “”;
echo “”;
echo “”;
echo “”;
echo “”;
echo “”;
echo “”;
echo “”;
echo ” “;
echo “”;
echo “”;
echo “”;
echo “”;
}
else
{
//memotong kata
function potong ($konten, $kata)
{
$TMPNEWSPART = array();
$TMP = explode(” “, $konten);
for($i=0; $i < $kata; $i++)
{
$TMPNEWSPART[$i] = $TMP[$i];
}
$NEWSPART = implode(" ", $TMPNEWSPART);
return $NEWSPART;
}
//memotong huruf
function potong_kata ($kata, $huruf_ke)
{
$potong_kata = substr($kata, $huruf_ke, 1);
return $potong_kata;
}
//mencari jumlah char
\\function panjang_kata ($kata)
{
$panjang_kata = strlen ($kata);
return $panjang_kata;
}
$kata1 = $_POST["kata1"];
$kata2 = $_POST["kata2"];
$kata_pertama = potong($kata1, 1);
$kata_kedua = potong($kata2, 1);
if ($kata1=='' or $kata2=='')
{
echo "Maaf, mungkin ada data yang belum anda masukkan!
klik disini untuk mengulangi.“;
}
else
{
echo “”;
echo “”;
echo “”;
echo “masukkan kata Pertama”;
echo “”;
echo “”;
echo “:”;
echo “”;
echo “”;
echo “$kata_pertama“;
echo “”;
echo “”;
echo “”;
echo “”;
echo “masukkan Kata Kedua “;
echo “”;
echo “”;
echo “:”;
echo “”;
echo “”;
echo “$kata_kedua“;
echo “”;
echo “”;
echo “”;
echo “”;
echo “Contanination”;
echo “”;
echo “”;
echo “:”;
echo “”;
echo “”;
echo “$kata_pertama$kata_kedua“;
echo “”;
echo “”;
echo “”;
echo “”;
echo “Shuffle”;
echo “”;
echo “”;
echo “:”;
echo “”;
echo “”;
echo ““;
$kata_pertama_mlai = 0;
$kata_kedua_mlai = 0;
$panjang_kata_1 = panjang_kata ($kata_pertama);
$panjang_kata_2 = panjang_kata ($kata_kedua);
if ( $panjang_kata_1 > $panjang_kata_2 )
{ $jml = $panjang_kata_1; }
else
{ $jml = $panjang_kata_2; }
for ( $x = 0; $x < $jml; $x++)
{
$kata_pertama_potong = potong_kata($kata_pertama, $kata_pertama_mlai);
$kata_kedua_potong = potong_kata($kata_kedua,$kata_kedua_mlai);
echo "$kata_pertama_potong";
echo "$kata_kedua_potong";
$kata_pertama_mlai = $kata_pertama_mlai + 1;
$kata_kedua_mlai = $kata_kedua_mlai + 1;
}
echo "“;
echo “”;
echo “”;
echo “”;
echo “”;
echo ” “;
echo “Untuk menjalankan program lagi klik disini.”;
echo “”;
echo “#################”;
echo “”;
echo “Terjadi kesalahan dalam menginputkan data..
maka kata pertama yang dipakai“;
echo “”;
}
}
echo “”;
echo “”;
?>
Nama: Tiyas Tri Kusumawati
NIM:10.11.4483
KELAS: S1 TI 1M
• 1. Example of antisymmetric matrix:
0 1 -2
A = -1 0 6
2 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
5 -2 8 0 -2 7
A = 2 0 3 B = 2 0 6
8 -2 4 -7 -6 0
5 2 8 0 2 -7
At = -2 0 -2Bt = -2 0 -6
8 3 4 7 6 0
-5 -2 -8 0 -2 7
-At =2 -0 2 -Bt = 2 0 6
-8 2 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama : Ade Gunawan Mohamad Zainal Putra
NIM : 10.11.4520
Kelas : S1-TI 1M
1. Example of antisymmetric matrix:
0 3 -2
A = -3 0 5
2 -5 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
An antisymmetric matrix (skew-symmetric matrix) is a square matrix A whose transpose is also its negative; that is, it satisfies the equation A = −AT.
Here are two examples of matrix, one with zero on each diagonal elements :
7 -4 9 0 -4 9
A = 4 2 8 B = 4 0 8
-9 -8 5 -9 -8 0
7 4 -9 0 4 -9
At = -4 2 -8 Bt = -4 0 -8
9 8 5 9 8 0
-7 -4 9 0 -4 9
-At = 4 -2 8 -Bt = 4 0 8
-9 -8 -5 -9 -8 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -3 7 0 11 -5
Let A = 3 0 -4 , B = -11 0 8
-7 4 0 5 -8 0
0+0 -3+11 7+(-5)
C = A + B = 3+(-11) 0+0 -4+8
-7+5 4+(-8) 0+0
0 8 2
= -8 0 4
-2 -4 0
0 -8 -2
Ct = 8 0 -4
2 4 0
0 8 2
-Ct = -8 0 4
-2 -4 0
C = -Ct (antisymmetric matrix)
Nama: Punditya Derry Dwi Pratikno
Nim: 10.11.4485
Kelas: 1M
a) the example of antisymmetric matrix
[0,7,2;7,0,-3;-2,3,0]
ans =
0 7 2
7 0 -3
-2 3 0
b) A= [0,-2,1;2,0,-3;-1,3,0]
A =
0 -2 1
2 0 -3
-1 3 0
>> B = [3,5,7;-4,2,-1;9,-2,6]
B =
3 5 7
-4 2 -1
9 -2 6
(A)transpose=
0 2 -1
-2 0 3
1 -3 0
(B)transpose=
3 -4 9
5 2 -2
7 -1 6
-(A)transpose=
0 -2 1
2 0 -3
-1 3 0
-(B)transpose=
-3 4 -9
-5 -2 2
-7 1 6
with that examples prove that -(A)transpose not equal to -(B)transpose
c) A= [0,-1,2;1,0,-4;-2,4,-0]
A =
0 -1 2
1 0 -4
-2 4 0
B= [0,-9,3;9,0,-7;-3,7,0]
B =
0 -9 3
9 0 -7
-3 7 0
[A+B]
ans =
0 -10 5
10 0 -11
-5 11 0
[A+B]transpose=
0 10 -5
-10 0 11
5 -11 0
-[A+B]transpose=
0 -10 5
10 0 -11
-5 11 0
in this case, prove that [A+B]=-[A+B]transpose
Nama : Yonathan Prasetioso.
NIM : 10.11.4517
Kelas : S1 TI 1M
a) Example of antisymmetric matrix:
>> [0,-4,8;4,0,-7;-8,7,0]
ans =
0 -4 8
4 0 -7
-8 7 0
b) Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A = [0,4,-2;-4,0,8;3,-8,0]
A =
0 4 -2
-4 0 8
3 -8 0
>> B = [6,-4,8;-5,1,6;2,-4,0]
B =
6 -4 8
-5 1 6
2 -4 0
>> (A)transpose
0 4 3
4 0 8
2 -8 0
>> (B)transpose
-6 5 -2
-4 -1 4
8 -6 0
>> -(A)transpose
0 -4 2
-4 0 -4
-2 8 0
>> -(B)transpose
6 -5 2
4 1 -4
-8 6 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
c) Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A = [0,4,9;-4,0,5;-9,-5,0]
A =
0 4 9
-4 0 5
-9 -5 0
>> B = [0,-3,8;3,0,-6;-8,6,0]
B =
0 -3 8
3 0 -6
-8 6 0
>> [A+B]
ans =
0 1 17
-1 0 -1
-17 1 0
>> [A+B]transpose
0 -1 -17
1 0 1
17 -1 0
>> -[A+B]transpose
0 1 17
-1 0 -1
17 1 0
Can be seen that the results of (A+B) equal to -(A+B)t.
Then it can be written (A+B) = – (A+B)t.
Nama : m. ulinnuha. n.h
Kelas : 1M
nim: 10.11.4544
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
-8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
Nama : muhammad ibrahim
Nim : 10.11.4481
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
Nama : Hermawan Agus Prasetyo
NIM : 10.11.4499
Kelas : 1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama : Jeri Wandana
NIM : 10.11.4526
1. Example of antisymmetric matrix
>> A = [0 1 -2;-1 0 -2;2 2 0]
A =
0 1 -2
-1 0 -2
2 2 0
>> -A
ans =
0 -1 2
1 0 2
-2 -2 0
>> -A’
ans =
0 1 -2
-1 0 -2
2 2 0
2. The sum of two antisymmetrics matrix
>> B = [0 2 -1;-2 0 -4;1 4 0]
B =
0 2 -1
-2 0 -4
1 4 0
>> A+B
ans =
0 3 -3
-3 0 -6
3 6 0
>> -(A+B)
ans =
0 -3 3
3 0 6
-3 -6 0
>> -(A+B)’
ans =
0 3 -3
-3 0 -6
3 6 0
3. Antisymmetric matrix is a square matrix having diagonal element zero
A =
0 1 -2
-1 0 -2
2 2 0
[0+0+0] = 0
Nama : Punditya Derry Dwi Pratikno
NIM : 10.11.4485
Kelas : S1 TI 1M
a) the example of antisymmetric matrix
[0,7,2;7,0,-3;-2,3,0]
ans =
0 7 2
7 0 -3
-2 3 0
b) A= [0,-2,1;2,0,-3;-1,3,0]
A =
0 -2 1
2 0 -3
-1 3 0
>> B = [3,5,7;-4,2,-1;9,-2,6]
B =
3 5 7
-4 2 -1
9 -2 6
(A)transpose=
0 2 -1
-2 0 3
1 -3 0
(B)transpose=
3 -4 9
5 2 -2
7 -1 6
-(A)transpose=
0 -2 1
2 0 -3
-1 3 0
-(B)transpose=
-3 4 -9
-5 -2 2
-7 1 6
with that examples prove that -(A)transpose not equal to -(B)transpose
c) A= [0,-1,2;1,0,-4;-2,4,-0]
A =
0 -1 2
1 0 -4
-2 4 0
B= [0,-9,3;9,0,-7;-3,7,0]
B =
0 -9 3
9 0 -7
-3 7 0
[A+B]
ans =
0 -10 5
10 0 -11
-5 11 0
[A+B]transpose=
0 10 -5
-10 0 11
5 -11 0
-[A+B]transpose=
0 -10 5
10 0 -11
-5 11 0
in this case, prove that [A+B]=-[A+B]transpose
Nama : Punditya Derry Dwi Pratikno
NIM : 10.11.4485
Kelas : S1 TI 1M
a) the example of antisymmetric matrix
[0,7,2;7,0,-3;-2,3,0]
ans =
0 7 2
7 0 -3
-2 3 0
b) A= [0,-2,1;2,0,-3;-1,3,0]
A =
0 -2 1
2 0 -3
-1 3 0
>> B = [3,5,7;-4,2,-1;9,-2,6]
B =
3 5 7
-4 2 -1
9 -2 6
(A)transpose=
0 2 -1
-2 0 3
1 -3 0
(B)transpose=
3 -4 9
5 2 -2
7 -1 6
-(A)transpose=
0 -2 1
2 0 -3
-1 3 0
-(B)transpose=
-3 4 -9
-5 -2 2
-7 1 6
with that examples prove that -(A)transpose not equal to -(B)transpose
c) A= [0,-1,2;1,0,-4;-2,4,-0]
A =
0 -1 2
1 0 -4
-2 4 0
B= [0,-9,3;9,0,-7;-3,7,0]
B =
0 -9 3
9 0 -7
-3 7 0
[A+B]
ans =
0 -10 5
10 0 -11
-5 11 0
[A+B]transpose=
0 10 -5
-10 0 11
5 -11 0
-[A+B]transpose=
0 -10 5
10 0 -11
-5 11 0
in this case, prove that [A+B]=-[A+B]transpose
Nama : M. ulinuha
NIM : 10.11.4544
Kelas : 1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
nama : rizal wahyudin
nim : 10.11.4534
kelas : s1 TI 1m
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa lihat bahwa A =-At dan B =-Bt. Jadi itu dapat di buktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti benar)
Nama : muhammad ibrahim
NIM : 10.11.4481
Kelas : 1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
nama : rizal wahyudin
nim : 10.11.4534
kelas : s1 TI 1m
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa lihat bahwa A =-At dan B =-Bt. Jadi itu dapat di buktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti benar)
Nama : m.ulinnuha
NIM : 10.11.4544
Kelas : 1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
Nama : M. Ulinuha N.H
NIM : 10.11.4544
Kelas : 1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
NAMA : RIZAL WAHYUDIN
NIM : 10.11.4534
KELAS : S1 TI 1M
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti benar)
NAMA : RIZAL WAHYUDIN
NIM : 10.11.4534
KELAS : S1 TI 1M
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti benar)
nama : deded fandilah
nim : 10.11.4515
kelas : S1 TI M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama: Rizky Aprio Prasetyo
NIM: 10.11.4496
Kelas: TI 1M
1. Example of antisymmetric matrix:
0 4 -2
A = -3 2 5
1 -5 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
6 -2 1 6 5 9
A = 4 2 5 B = -2 0 -1
-8 -3 2 1 -4 2
-6 2 1 -6 -5 -9
At = -4 -2 -5 Bt = 2 0 1
8 3 -2 -1 4 -2
6 -2 1 6 5 9
-At = 4 2 5 -Bt = -2 0 -1
-8 -3 2 1 -4 2
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
1 2 1 4 -2 -5
Let A = 3 0 3 , B = -1 0 4
-2 3 0 5 -4 0
1+4 2+(-2) 1+(-5)
C = A + B = 3+(-1) 0+0 3+4
-2+5 3+(-4) 0+0
5 0 -4
= 2 0 7
3 -1 0
5 0 -4
Ct = 2 0 7
3 -1 0
-5 0 4
-Ct = -2 0 -7
-3 1 0
C = -Ct (antisymmetric matrix)
Nama : Ibnu Priharyanto
NIM : 10.21.0489
1).jawaban soal A
>> A=[5 6 7;7 6 5;6 7 5]
A =
5 6 7
7 6 5
6 7 5
>> -A’
ans =
-5 -7 -6
-6 -6 -7
-7 -5 -5
2).Jawaban Soal B
>>A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3).Jawaban Soal C
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama :Ngadiyanto
NIM : 10.21.0487
1).jawaban soal A
>> A=[7 8 9;9 8 7;8 9 7]
A =
7 8 9
9 8 7
8 9 7
>> -A’
ans =
-7 -9 -8
-8 -7 -9
-9 -7 -7
2).Jawaban Soal B
>>A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3).Jawaban Soal C
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama : Oky Klinston Tambun
NIM : 10.11.4494
Kelas : S1 TI 1M
1. Example of antisymmetric matrix:
A = 0 1 5
-1 0 -6
-5 6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
A = 0 5 -9
-5 0 3
9 -3 0
At = 0 -5 9
5 0 -3
-9 3 0
-At = 0 5 -9
-5 0 3
-9 -3 0
B = 5 0 4
-6 2 7
2 3 8
Bt = 5 -6 2
0 2 3
4 7 8
-Bt = -5 6 -2
0 -2 -3
-4 -7 -8
Terbukti bahwa -At mempunyai diagonal nol. Maka antisymetric -At mempunyai diagonal matrik.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
A = 0 1 -3
-1 0 6
3 -6 0
B = 0 -2 4
2 0 -7
-4 7 0
C = 0 -1 1
1 0 -1
-1 1 0
Ct = 0 1 -1
-1 0 1
1 -1 0
-Ct = 0 -1 1
1 0 -1
-1 1 0
terbukti C = – Ct.
nama : febrian budi utama
nim : 10.11.4543
kls : S1 TI 1M
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti benar)
NAMA : Pandu Yudistira Nugraha
NIM : 10.11.4508
KELAS : S1 TI 1M
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti benar)
Nama : Yonathan Prasetioso.
NIM : 10.11.4517
Kelas : S1 TI 1M
1.) A matrix A is said to be antisymmetric if A : -At.
a.) Example of antisymmetric matrix:
A = 0 -4 8
4 0 -7
-8 7 0
b.) Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
A = 0 4 -2 B = 6 -4 8
-4 0 8 -5 1 6
3 -8 0 2 -4 0
At = 0 -4 2 Bt = -6 4 -8
4 0 -8 5 -1 -6
-3 8 0 -2 4 0
-At = 0 4 -2 -Bt = 6 -4 8
-4 0 8 -5 1 6
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
c.) Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
A = 0 4 9 B = 0 -3 8
-4 0 5 3 0 -6
-9 -5 0 -8 6 0
(A+B) = 0 1 17
-1 0 -1
-17 1 0
(A+B)t = 0 -1 -17 -(A+B)t = 0 1 17
1 0 1 -1 0 -1
17 -1 0 -17 1 0
Can be seen that the results of (A+B) equal to -(A+B)t.
Then it can be written (A+B) = – (A+B)t.
NAMA : SURYAWAN PRIANTORO
NIM : 10.11.4533
KELAS: S1-TI 1M
1.) Example of antisymmetric matrix:
0 4 -2
A = -3 2 5
1 -5 0
2.) Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
6 -2 1 6 5 9
A = 4 2 5 B = -2 0 -1
-8 -3 2 1 -4 2
-6 2 1 -6 -5 -9
At = -4 -2 -5 Bt = 2 0 1
8 3 -2 -1 4 -2
6 -2 1 6 5 9
-At = 4 2 5 -Bt = -2 0 -1
-8 -3 2 1 -4 2
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.*
3.) Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
1 2 1 4 -2 -5
Let A = 3 0 3 , B = -1 0 4
-2 3 0 5 -4 0
1+4 2+(-2) 1+(-5)
C = A + B = 3+(-1) 0+0 3+4
-2+5 3+(-4) 0+0
5 0 -4
= 2 0 7
3 -1 0
5 0 -4
Ct = 2 0 7
3 -1 0
-5 0 4
-Ct = -2 0 -7
-3 1 0
C = -Ct (antisymmetric matrix)*
1. Example of antisymmetric matrix:
5 4 2
A = -1 3 0
-1 3 -2
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
3 1 0 -3 2 6
A = -2 0 1 B = 2 -1 0
1 2 1 3 -2 1
-3 -1 0 3 -2 -6
At = 2 0 -1 Bt = -2 1 0
-1 -2 -1 -3 2 -1
3 1 0 -3 2 6
-At = -2 0 1 -Bt = 2 -1 0
1 2 1 3 -2 1
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
6 2 1 -3 2 0
Let A = 1 -2 1 , B = 2 -2 1
0 2 -1 3 2 0
6+(-3) 2+2 1+0
C = A + B = 1+2 -2+(-2) 1+1
0+3 2+2 -1+0
3 4 1
= 3 0 2
3 4 0
-3 -4 -1
Ct = -3 0 -2
-3 -4 0
3 4 1
-Ct = 3 0 2
3 4 0
C = -Ct (antisymmetric matrix)
Nama : Wahyu Adi Setyawan
Kelas : S1 TI-M
NIM : 10.11.4478
1. Example of antisymmetric matrix:
0 4 -2
A = -3 2 5
1 -5 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
6 -2 1 6 5 9
A = 4 2 5 B = -2 0 -1
-8 -3 2 1 -4 2
-6 2 1 -6 -5 -9
At = -4 -2 -5 Bt = 2 0 1
8 3 -2 -1 4 -2
6 -2 1 6 5 9
-At = 4 2 5 -Bt = -2 0 -1
-8 -3 2 1 -4 2
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
1 2 1 4 -2 -5
Let A = 3 0 3 , B = -1 0 4
-2 3 0 5 -4 0
1+4 2+(-2) 1+(-5)
C = A + B = 3+(-1) 0+0 3+4
-2+5 3+(-4) 0+0
5 0 -4
= 2 0 7
3 -1 0
5 0 -4
Ct = 2 0 7
3 -1 0
-5 0 4
-Ct = -2 0 -7
-3 1 0
C = -Ct (antisymmetric matrix)
Nama : Zakiul fahmi
Nmor Mhsiswa : 10.11.4537
Kelas : S1-TI-1M
A)
A= 0 -4 -6
4 0 -1
6 1 0
B)
A= 0 -2 5
At = 0 2 -5
-At = 0 -2 5
2 0 -3 -2 0 3 2 0 -3
-5 3 0 5 -3 2 -5 3 0
( 0 miring ke bawah sebagai diagonal Matrik )
A = 1 -3 5
Bt = 1 0 -4
-Bt = -1 0 4
0 6 -2 3 6 0 3 -6 0
-4 0 8 5 -2 8 -5 2 -8
( kesimpulan : -At dan -Bt tidak sama
-At mempunyai diagonal bernilai nol, maka terbukti bahwa arsymetrik -At mempunyai diagonal matriks )
C)
A = 0 -9 1
b = 0 5 2 (a+b) = 0 -4 3
9 0 -3 -5 0 1 4 0 -2
-1 3 0 -2 -1 0 -3 2 0 ( sama )
(A+B)t : 0 4 -3 -(A+B) : 0 4 3
-4 0 2 4 0 -2
3 -2 0 -3 2 0
( terbukti bahwa (A+B) = -(A+B)t (antysimatrik matriks) )
Pembuktian dengan menggunakan Matlab 7.10
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
NAMA : Yusiana Pambaruni
NIM : 10.11.4497
KELAS : S1 TI 1M
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti benar)
NAMA : Gani Anggoro
NIM : 10.11.4510
KELAS : S1 TI 1M
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti benar)
NAMA : AKHMAD NUR FAOSIN
NIM : 10.11.4509
KELAS : S1 TI 1M
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti benar)
Nama : Helmy Ashari
Nim : 10.11.4535
Kls : S1 TI 1M
a)
A= 0 -4 -6
4 0 -1
6 1 0
b)
A= 0 -2 5
At = 0 2 -5
-At = 0 -2 5
2 0 -3 -2 0 3 2 0 -3
-5 3 0 5 -3 2 -5 3 0
( 0 miring ke bawah sebagai diagonal Matrik )
A = 1 -3 5
Bt = 1 0 -4
-Bt = -1 0 4
0 6 -2 3 6 0 3 -6 0
-4 0 8 5 -2 8 -5 2 -8
( kesimpulan : -At dan -Bt tidak sama
-At mempunyai diagonal bernilai nol, maka terbukti bahwa arsymetrik -At mempunyai diagonal matriks )
c)
A = 0 -9 1
b = 0 5 2 (a+b) = 0 -4 3
9 0 -3 -5 0 1 4 0 -2
-1 3 0 -2 -1 0 -3 2 0 ( sama )
(A+B)t : 0 4 -3 -(A+B) : 0 4 3
-4 0 2 4 0 -2
3 -2 0 -3 2 0
( terbukti bahwa (A+B) = -(A+B)t (antysimatrik matriks) )
Nama : Nurul Qarama
NIM : 10.11.4498
Kelas : S1-TI 1M
1. Example of antisymmetric matrix:
0 4 -1
A = -4 0 7
1 -7 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
An antisymmetric matrix (skew-symmetric matrix) is a square matrix A whose transpose is also its negative; that is, it satisfies the equation A = −AT.
Here are two examples of matrix, one with zero on each diagonal elements :
6 -5 8 0 -5 8
A = 3 1 9 B = 3 0 9
-8 -9 6 -8 -9 0
6 5 -8 0 -5 -8
At = -3 1 -9 Bt = -3 0 -9
8 9 6 8 9 0
6 -5 8 0 -5 8
-At = 3 1 9 -Bt = 3 0 9
-8 -9 6 -8 -9 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -5 9 0 10 -6
Let A = 5 0 -3 , B = -10 0 7
-9 3 0 6 -7 0
0+0 -5+10 9+(-6)
C = A + B = 5+(-10) 0+0 -3+7
-9+6 3+(-7) 0+0
0 5 3
= -5 0 4
-3 -4 0
0 -5 -3
Ct = 5 0 -4
3 4 0
0 5 3
-Ct = -5 0 4
-3 -4 0
C = -Ct (antisymmetric matrix)
Nama : Nurul Qarama
NIM : 10.11.4498
Kelas : S1-TI 1M
1. Example of antisymmetric matrix:
0 4 -1
A = -4 0 7
1 -7 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
An antisymmetric matrix (skew-symmetric matrix) is a square matrix A whose transpose is also its negative; that is, it satisfies the equation A = −AT.
Here are two examples of matrix, one with zero on each diagonal elements :
6 -5 8 0 -5 8
A = 3 1 9 B = 3 0 9
-8 -9 6 -8 -9 0
6 5 -8 0 -5 -8
At = -3 1 -9 Bt = -3 0 -9
8 9 6 8 9 0
6 -5 8 0 -5 8
-At = 3 1 9 -Bt = 3 0 9
-8 -9 6 -8 -9 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -5 9 0 10 -6
Let A = 5 0 -3 , B = -10 0 7
-9 3 0 6 -7 0
0+0 -5+10 9+(-6)
C = A + B = 5+(-10) 0+0 -3+7
-9+6 3+(-7) 0+0
0 5 3
= -5 0 4
-3 -4 0
0 -5 -3
Ct = 5 0 -4
3 4 0
0 5 3
-Ct = -5 0 4
-3 -4 0
C = -Ct (antisymmetric matrix)
NIM : 10.11.4519
Nama : Topan Aji Nugroho
1. Example of antisymmetric matrix:
5 4 2
A = -1 3 0
-1 3 -2
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
3 1 0 -3 2 6
A = -2 0 1 B = 2 -1 0
1 2 1 3 -2 1
-3 -1 0 3 -2 -6
At = 2 0 -1 Bt = -2 1 0
-1 -2 -1 -3 2 -1
3 1 0 -3 2 6
-At = -2 0 1 -Bt = 2 -1 0
1 2 1 3 -2 1
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
6 2 1 -3 2 0
Let A = 1 -2 1 , B = 2 -2 1
0 2 -1 3 2 0
6+(-3) 2+2 1+0
C = A + B = 1+2 -2+(-2) 1+1
0+3 2+2 -1+0
3 4 1
= 3 0 2
3 4 0
-3 -4 -1
Ct = -3 0 -2
-3 -4 0
3 4 1
-Ct = 3 0 2
3 4 0
C = -Ct (antisymmetric matrix)
NIM : 10.11.4519
Nama : Topan Aji Nugroho
1. Example of antisymmetric matrix:
5 4 2
A = -1 3 0
-1 3 -2
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
3 1 0 -3 2 6
A = -2 0 1 B = 2 -1 0
1 2 1 3 -2 1
-3 -1 0 3 -2 -6
At = 2 0 -1 Bt = -2 1 0
-1 -2 -1 -3 2 -1
3 1 0 -3 2 6
-At = -2 0 1 -Bt = 2 -1 0
1 2 1 3 -2 1
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
6 2 1 -3 2 0
Let A = 1 -2 1 , B = 2 -2 1
0 2 -1 3 2 0
6+(-3) 2+2 1+0
C = A + B = 1+2 -2+(-2) 1+1
0+3 2+2 -1+0
3 4 1
= 3 0 2
3 4 0
-3 -4 -1
Ct = -3 0 -2
-3 -4 0
3 4 1
-Ct = 3 0 2
3 4 0
C = -Ct (antisymmetric matrix)
Nama : Nurul Qarama
NIM : 10.11.4498
kelas : S1-TI 1M
1. Example of antisymmetric matrix:
0 4 -1
A = -4 0 7
1 -7 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
An antisymmetric matrix (skew-symmetric matrix) is a square matrix A whose transpose is also its negative; that is, it satisfies the equation A = −AT.
Here are two examples of matrix, one with zero on each diagonal elements :
6 -5 8 0 -5 8
A = 3 1 9 B = 3 0 9
-8 -9 6 -8 -9 0
6 5 -8 0 -5 -8
At = -3 1 -9 Bt = -3 0 -9
8 9 6 8 9 0
6 -5 8 0 -5 8
-At = 3 1 9 -Bt = 3 0 9
-8 -9 6 -8 -9 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -5 9 0 10 -6
Let A = 5 0 -3 , B = -10 0 7
-9 3 0 6 -7 0
0+0 -5+10 9+(-6)
C = A + B = 5+(-10) 0+0 -3+7
-9+6 3+(-7) 0+0
0 5 3
= -5 0 4
-3 -4 0
0 -5 -3
Ct = 5 0 -4
3 4 0
0 5 3
-Ct = -5 0 4
-3 -4 0
C = -Ct (antisymmetric matrix)
NIM : 10.11.4483
Nama : Tiyas Tri Kusumawati
Kelas : S1 TI -M
1. Example of antisymmetric matrix:
2 -1 2
A = 2 3 0
1 -4 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
2 4 0 2 -1 0
A = -2 -1 0 B = 0 2 -5
-2 -1 3 2 2 -1
-2 -4 0 -2 1 0
At = 2 1 0 Bt = 0 -2 5
2 1 -3 -2 -2 1
2 4 0 2 -1 0
-At = -2 -1 0 -Bt = 0 -2 5
-2 -1 3 2 2 -1
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
2 -3 1 3 -4 2
Let A = 1 -1 0 , B = 1 -2 3
0 2 1 4 4 5
2+3 -3+(-4) 1+2
C = A + B = 1+1 -1+(-2) 0+3
0+4 2+4 1+5
5 -7 3
= 2 -3 3
4 6 6
-5 7 -3
Ct = -2 3 -3
-4 -6 -6
5 -7 3
-Ct = 2 -3 3
4 6 6
C = -Ct (antisymmetric matrix)
Nama : Ade Gunawan Mohamad Zainal Putra
NIM : 10.11.4520
Kelas : S1-TI 1M
1. Example of antisymmetric matrix:
0 3 -2
A = -3 0 5
2 -5 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
An antisymmetric matrix (skew-symmetric matrix) is a square matrix A whose transpose is also its negative; that is, it satisfies the equation A = −At.
Here are two examples of matrix, one with zero on each diagonal elements :
7 -4 9 0 -4 9
A = 4 2 8 B = 4 0 8
-9 -8 5 -9 -8 0
7 4 -9 0 4 -9
At = -4 2 -8 Bt = -4 0 -8
9 8 5 9 8 0
-7 -4 9 0 -4 9
-At = 4 -2 8 -Bt = 4 0 8
-9 -8 -5 -9 -8 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -3 7 0 11 -5
Let A = 3 0 -4 , B = -11 0 8
-7 4 0 5 -8 0
0+0 -3+11 7+(-5)
C = A + B = 3+(-11) 0+0 -4+8
-7+5 4+(-8) 0+0
0 8 2
= -8 0 4
-2 -4 0
0 -8 -2
Ct = 8 0 -4
2 4 0
0 8 2
-Ct = -8 0 4
-2 -4 0
C = -Ct (antisymmetric matrix) Proved!
NIM : 10.11.4518
Nama : Sinta Sulistyaningsih
1. Example of antisymmetric matrix:
1 1 0
A = -3 2 1
1 -4 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
1 2 1 2 1 -2
A = -1 0 2 B = -1 -3 0
-2 -1 3 2 -1 3
-1 -2 -1 -2 -1 2
At = -1 0 -2 Bt = 1 3 0
2 1 -3 -2 1 -3
1 2 1 2 1 -2
-At = -1 0 2 -Bt = -1 -3 0
-2 -1 3 2 -1 3
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
1 2 -2 0 -1 3
Let A = 3 1 -2 , B = 2 -2 1
3 1 1 4 2 1
1+0 2+(-1) -2+3
C = A + B = 3+2 1+(-2) -2+1
3+4 1+2 1+1
1 1 1
= 5 -1 -1
7 3 2
-1 -1 -1
Ct = -5 1 1
-7 -3 -2
1 1 1
-Ct = 5 -1 -1
7 3 2
C = -Ct (antisymmetric matrix)
NIM : 10.11.4482
Nama : Listiyaningsih
Kelas : S1 TI M
1. Example of antisymmetric matrix:
1 3 -3
A = -3 2 2
1 -4 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
3 -1 0 3 3 2
A = 2 2 1 B = -2 0 4
-2 -1 3 1 1 -3
-3 1 0 -3 -3 -2
At = -2 -2 -1 Bt = 2 0 -4
2 1 -3 -1 -1 3
3 -1 0 3 3 2
-At = 2 2 1 -Bt = -2 0 4
-2 -1 3 1 1 -3
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
2 1 -3 2 2 -4
Let A = 0 1 -2 , B = 1 -3 1
3 -1 3 4 2 1
2+2 1+2 -3+(-4)
C = A + B = 0+1 1+(-3) -2+1
3+4 -1+2 3+1
4 3 -7
= 1 -5 -1
7 1 4
-4 -3 7
Ct = -1 5 1
-7 -1 -4
4 3 -7
-Ct = 1 -5 -1
7 1 4
C = -Ct (antisymmetric matrix)
Nama : Lisman prihadi
Nim : 10.11.4493
Kls : S1 TI 1M
a) A= 0 -4 -6
4 0 -1
6 1 0
b) A= 0 -2 5 At = 0 2 -5 -At = 0 -2 5
2 0 -3 -2 0 3 2 0 -3
-5 3 0 5 -3 2 -5 3 0
0 miring ke bawah sebagai diagonal Matrik
A = 1 -3 5 Bt = 1 0 -4 -Bt = -1 0 4
0 6 -2 3 6 0 3 -6 0
-4 0 8 5 -2 8 -5 2 -8
kesimpulan : -At dan -Bt tidak sama
-At mempunyai diagonal bernilai nol, maka terbukti bahwa arsymetrik -At mempunyai diagonal matriks
c) A = 0 -9 1 b = 0 5 2 (a+b) = 0 -4 3
9 0 -3 -5 0 1 4 0 -2
-1 3 0 -2 -1 0 -3 2 0 sama
(A+B)t : 0 4 -3 -(A+B) : 0 4 3
-4 0 2 4 0 -2
3 -2 0 -3 2 0
terbukti bahwa (A+B) = -(A+B)t (antysimatrik matriks)
Nim : 4491
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nim : 4491
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama : Lisman Prihadi
NIM : 10.11.4493
Kelas : S1 TI M
1. Example of antisymmetric matrix:
0 4 -2
A = -3 2 5
1 -5 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
6 -2 1 6 5 9
A = 4 2 5 B = -2 0 -1
-8 -3 2 1 -4 2
-6 2 1 -6 -5 -9
At = -4 -2 -5 Bt = 2 0 1
8 3 -2 -1 4 -2
6 -2 1 6 5 9
-At = 4 2 5 -Bt = -2 0 -1
-8 -3 2 1 -4 2
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
1 2 1 4 -2 -5
Let A = 3 0 3 , B = -1 0 4
-2 3 0 5 -4 0
1+4 2+(-2) 1+(-5)
C = A + B = 3+(-1) 0+0 3+4
-2+5 3+(-4) 0+0
5 0 -4
= 2 0 7
3 -1 0
5 0 -4
Ct = 2 0 7
3 -1 0
-5 0 4
-Ct = -2 0 -7
-3 1 0
C = -Ct (antisymmetric matrix)
NAMA : FENDI PRIYANTO
NIM : 10.11.4488
KELAS : SI TI / M
Dengan menggunakan matlab 7.10
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct antisymmetric matrix berarti benar
Nama : Doni Rusman Teja
NIM : 10.11.4492
Kelas : S1 TI M
1. Example of antisymmetric matrix:
0 4 -2
A = -3 2 5
1 -5 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
6 -2 1 6 5 9
A = 4 2 5 B = -2 0 -1
-8 -3 2 1 -4 2
-6 2 1 -6 -5 -9
At = -4 -2 -5 Bt = 2 0 1
8 3 -2 -1 4 -2
6 -2 1 6 5 9
-At = 4 2 5 -Bt = -2 0 -1
-8 -3 2 1 -4 2
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
1 2 1 4 -2 -5
Let A = 3 0 3 , B = -1 0 4
-2 3 0 5 -4 0
1+4 2+(-2) 1+(-5)
C = A + B = 3+(-1) 0+0 3+4
-2+5 3+(-4) 0+0
5 0 -4
= 2 0 7
3 -1 0
5 0 -4
Ct = 2 0 7
3 -1 0
-5 0 4
-Ct = -2 0 -7
-3 1 0
C = -Ct (antisymmetric matrix)
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
7 -5 7 0 -5 7
A = 5 2 8 B = 5 0 8
-7 -8 5 -7 -8 0
7 5 -7 0 5 -7
At = -5 2 -8 Bt = -5 0 -8
7 8 5 7 8 0
-7 -5 7 0 -5 7
-At = 5 -2 8 -Bt = 5 0 8
-7 -8 -5 -7 -8 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -2 4 0 7 -5
Let A = 2 0 -6 , B = -7 0 3
-4 6 0 5 -3 0
C = A + B
A B C
0 -2 4 0 7 -5 0 5 -1
2 0 -6 + -7 0 3 = -5 0 -3
-4 6 0 5 -3 0 1 3 0
Ct = 0 -5 -1
5 0 3
-1 -3 0
-Ct = 0 5 -1
-5 0 -3
1 3 0
C = -Ct (antisymmetric matrix)
So A+B dapat dituliskan -(A+B)t
Nama : Dyta Romania
Kelas : 1M
NIM : 10.11.4500
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
7 -5 7 0 -5 7
A = 5 2 8 B = 5 0 8
-7 -8 5 -7 -8 0
7 5 -7 0 5 -7
At = -5 2 -8 Bt = -5 0 -8
7 8 5 7 8 0
-7 -5 7 0 -5 7
-At = 5 -2 8 -Bt = 5 0 8
-7 -8 -5 -7 -8 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -2 4 0 7 -5
Let A = 2 0 -6 , B = -7 0 3
-4 6 0 5 -3 0
C = A + B
A B C
0 -2 4 0 7 -5 0 5 -1
2 0 -6 + -7 0 3 = -5 0 -3
-4 6 0 5 -3 0 1 3 0
Ct = 0 -5 -1
5 0 3
-1 -3 0
-Ct = 0 5 -1
-5 0 -3
1 3 0
C = -Ct (antisymmetric matrix)
So A+B dapat dituliskan -(A+B)t
Nama : Lisman Prihadi
NIM : 10.11.4493
Kelas : S1 TI M
1. Example of antisymmetric matrix:
0 4 -2
A = -3 2 5
1 -5 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
6 -2 1 6 5 9
A = 4 2 5 B = -2 0 -1
-8 -3 2 1 -4 2
-6 2 1 -6 -5 -9
At = -4 -2 -5 Bt = 2 0 1
8 3 -2 -1 4 -2
6 -2 1 6 5 9
-At = 4 2 5 -Bt = -2 0 -1
-8 -3 2 1 -4 2
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
1 2 1 4 -2 -5
Let A = 3 0 3 , B = -1 0 4
-2 3 0 5 -4 0
1+4 2+(-2) 1+(-5)
C = A + B = 3+(-1) 0+0 3+4
-2+5 3+(-4) 0+0
5 0 -4
= 2 0 7
3 -1 0
5 0 -4
Ct = 2 0 7
3 -1 0
-5 0 4
-Ct = -2 0 -7
-3 1 0
C = -Ct (antisymmetric matrix)
Nama : Lisman Prihadi
NIM : 10.11.4493
Kelas : S1 TI M
1. Example of antisymmetric matrix:
0 4 -2
A = -3 2 5
1 -5 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
6 -2 1 6 5 9
A = 4 2 5 B = -2 0 -1
-8 -3 2 1 -4 2
-6 2 1 -6 -5 -9
At = -4 -2 -5 Bt = 2 0 1
8 3 -2 -1 4 -2
6 -2 1 6 5 9
-At = 4 2 5 -Bt = -2 0 -1
-8 -3 2 1 -4 2
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
1 2 1 4 -2 -5
Let A = 3 0 3 , B = -1 0 4
-2 3 0 5 -4 0
1+4 2+(-2) 1+(-5)
C = A + B = 3+(-1) 0+0 3+4
-2+5 3+(-4) 0+0
5 0 -4
= 2 0 7
3 -1 0
5 0 -4
Ct = 2 0 7
3 -1 0
-5 0 4
-Ct = -2 0 -7
-3 1 0
C = -Ct (antisymmetric matrix)
Nama : MUHAMMAD YUSUF
Kelas : S1 TI-1M
NIM : 10.11.4469
1. Example of antisymmetric matrix:
0 4 -2
A = -3 2 5
1 -5 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
6 -2 1 6 5 9
A = 4 2 5 B = -2 0 -1
-8 -3 2 1 -4 2
-6 2 1 -6 -5 -9
At = -4 -2 -5 Bt = 2 0 1
8 3 -2 -1 4 -2
6 -2 1 6 5 9
-At = 4 2 5 -Bt = -2 0 -1
-8 -3 2 1 -4 2
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
1 2 1 4 -2 -5
Let A = 3 0 3 , B = -1 0 4
-2 3 0 5 -4 0
1+4 2+(-2) 1+(-5)
C = A + B = 3+(-1) 0+0 3+4
-2+5 3+(-4) 0+0
5 0 -4
= 2 0 7
3 -1 0
5 0 -4
Ct = 2 0 7
3 -1 0
-5 0 4
-Ct = -2 0 -7
-3 1 0
C = -Ct (antisymmetric matrix)
1. Example of antisymmetric matrix:
A=[1 2 1;2 0 3; 2 2 0]
A =
1 2 1
2 0 3
2 2 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[0 1 1;2 3 1; 2 1 2],B=[2 1 1; 2 3 1;2 1 0]
A =
0 1 1
2 3 1
2 1 2
B =
2 1 1
2 3 1
2 1 0
>> A’,B’
ans =
0 2 2
1 3 1
1 1 2
ans =
2 2 2
1 3 1
1 1 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[1 2 4; 0 2 1;1 2 4],B=[1 2 3;1 0 1;2 3 1]
A =
1 2 4
0 2 1
1 2 4
B =
1 2 3
1 0 1
2 3 1
>> C=A+B
C =
2 4 7
1 2 2
3 5 5
>> C’
ans =
2 1 3
4 2 5
7 2 5
>> -C’
ans =
-2 -1 -3
-4 -2 -5
-7 -2 -5
Hasil dari (A+B) samadengan -(A+B)t.
Maka dapat dituliskan (A+B) = – (A+B)t.
Nama : Hampron Triandoko
NIM : 10.11.4529
Kelas : S1 TI 1M
1. Example of antisymmetric matrix:
0 4 -2
A = -3 2 5
1 -5 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
6 -2 1 6 5 9
A = 4 2 5 B = -2 0 -1
-8 -3 2 1 -4 2
-6 2 1 -6 -5 -9
At = -4 -2 -5 Bt = 2 0 1
8 3 -2 -1 4 -2
6 -2 1 6 5 9
-At = 4 2 5 -Bt = -2 0 -1
-8 -3 2 1 -4 2
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
1 2 1 4 -2 -5
Let A = 3 0 3 , B = -1 0 4
-2 3 0 5 -4 0
1+4 2+(-2) 1+(-5)
C = A + B = 3+(-1) 0+0 3+4
-2+5 3+(-4) 0+0
5 0 -4
= 2 0 7
3 -1 0
5 0 -4
Ct = 2 0 7
3 -1 0
-5 0 4
-Ct = -2 0 -7
-3 1 0
C = -Ct (antisymmetric matrix)
1. Example of antisymmetric matrix:
A=[1 2 3;4 0 3;1 2 0]
A =
1 2 3
4 0 3
1 2 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[0 2 3;1 2 1;3 2 1],B=[2 0 1;2 1 1; 0 3 2]
A =
0 2 3
1 2 1
3 2 1
B =
2 0 1
2 1 1
0 3 2
>> A’,B’
ans =
0 1 3
2 2 2
3 1 1
ans =
2 2 0
0 1 3
1 1 2
>> -A’,-B’
ans =
0 -1 -3
-2 -2 -2
-3 -1 -1
ans =
-2 -2 0
0 -1 -3
-1 -1 -2
A =-At dan B =-Bt. Jadi membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[1 2 3;4 3 1;0 2 1],B=[1 2 3;0 3 2;1 4 1]
A =
1 2 3
4 3 1
0 2 1
B =
1 2 3
0 3 2
1 4 1
>> C=A+B
C =
2 4 6
4 6 3
1 6 2
>> C’
ans =
2 4 1
4 6 6
6 3 2
>> -C’
ans =
-2 -4 -1
-4 -6 -6
-6 -3 -2
Hasil dari (A+B) samadengan -(A+B)t.
Maka dapat dituliskan (A+B) = – (A+B)t.
Nama : Dyta Romania
NIM : 10.11.4500
kelas : 1M
1.Example of antisymmetric matrix:
A = 0 5 -2
-3 0 -5
2 3 0
2.Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
A = 1 3 5 B = 0 3 5
-3 2 7 -3 0 7
-5 -7 4 -5 -7 0
At = 1 -3 -5 Bt = 0 -3 -5
3 2 -7 3 0 -7
5 7 4 5 7 0
-At = -1 3 5 -Bt = 0 3 5
3 -2 7 -3 0 7
-5 -7 -4 -5 -7 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3.Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
Let A = 0 2 -5 Let B = 0 -3 -5
-2 0 -3 3 0 9
5 3 0 5 -9 0
C = A+B
A B C
0 2 -5 0 -3 -5 0 -1 -10
-2 0 -3 + 3 0 9 = 1 0 6
5 3 0 5 -9 0 10 -6 0
Ct = 0 1 10
-1 0 -6
-10 6 0
-Ct = 0 -1 -10
1 0 6
10 -6 0
So C= -Ct merupakan antisymmetric matrix…. terbukti….///>
Nama : Rakhmat Solihin
NIM : 10.11.4507
Kelas : S1.TI.1M
1. Example of antisymmetric matrix:
0 3 -2
A = -3 0 1
2 -1 0
2. Prove that antisymmetric matrix is a square matrix with
zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
8 -4 9 0 -4 9
A = 4 5 6 B = 4 0 6
-9 -6 3 -9 -6 0
8 4 -9 0 4 -9
At = -4 5 -6 Bt = -4 0 -6
9 6 3 9 6 0
-8 -4 9 0 -4 9
-At = 4 -5 6 -Bt = 4 0 6
-9 -6 -3 -9 -6 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an
antisymmetric matrix.
0 -4 9 0 3 -8
A = 4 0 -6 , B = -3 0 7
-9 6 0 8 -7 0
0+0 -4+3 9+(-8)
C = A + B = 4+(-3) 0+0 -6+7
-9+8 6+(-7) 0+0
0 -1 1
= 1 0 1
-1 -1 0
0 1 -1
Ct = -1 0 -1
1 1 0
0 -1 1
-Ct = 1 0 1
-1 -1 0
C = -Ct (antisymmetric matrix)
Nama
1.Example of antisymmetric matrix:
A = 0 5 -2
-3 0 -5
2 3 0
2.Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
A = 1 3 5 B = 0 3 5
-3 2 7 -3 0 7
-5 -7 4 -5 -7 0
At = 1 -3 -5 Bt = 0 -3 -5
3 2 -7 3 0 -7
5 7 4 5 7 0
-At = -1 3 5 -Bt = 0 3 5
-3 -2 7 -3 0 7
-5 -7 -4 -5 -7 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3.Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
Let A = 0 2 -5 Let B = 0 -3 -5
-2 0 -3 3 0 9
5 3 0 5 -9 0
C = A+B
A B C
0 2 -5 0 -3 -5 0 -1 -10
-2 0 -3 + 3 0 9 = 1 0 6
5 3 0 5 -9 0 10 -6 0
Ct = 0 1 10
-1 0 -6
-10 6 0
-Ct = 0 -1 -10
1 0 6
10 -6 0
So C = – Ct antisyimmetric matric True…
NIM: 10.11.4475
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti benar)
Nama : Dian Nur A.
NIM : 10.11.4516
Kelas : S1 TI 1M
1. Example of antisymmetric matrix:
A = 0 -5 9
5 0 -7
-9 7 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
A = 0 2 -1
-2 0 7
1 -7 0
At = 0 -2 1
2 0 -7
-1 7 0
-At = 0 2 -1
-2 0 7
1 -7 0
B = 4 -2 7
-3 1 5
6 -8 0
Bt = 4 -3 6
-2 1 -8
7 5 0
-Bt = -4 3 -6
2 -1 8
-7 -5 0
dari pembuktian di atas, -At dan -Bt berbeda. Terbukti bahwa -At mempunyai diagonal nol. Maka antisymetric -At mempunyai diagonal matrik.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
A = 0 4 9
-4 0 5
-9 -5 0
B = 0 -3 8
3 0 -6
-8 6 0
(A+B) = 0 1 17
-1 0 -1
-17 1 0
(A+B)t = 0 -1 -17
1 0 1
17 -1 0
-(A+B)t = 0 1 17
-1 0 -1
-17 1 0
Dapat dilihat bahwa hasil dari (A+B) sama dengan -(A+B)t.
Maka dapat dituliskan (A+B) = – (A+B)t.
Nama : Dian Nur A.
NIM : 10.11.4516
Kelas : S1 TI 1M
1. Example of antisymmetric matrix:
A = 0 -5 9
5 0 -7
-9 7 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
A = 0 2 -1 B = 4 -2 7
-2 0 7 -3 1 5
1 -7 0 6 -8 0
At = 0 -2 1 Bt = 4 -3 6
2 0 -7 -2 1 -8
-1 7 0 7 5 0
-At = 0 2 -1 -Bt = -4 3 -6
-2 0 7 2 -1 8
1 -7 0 -7 -5 0
dari pembuktian di atas, -At dan -Bt berbeda. Terbukti bahwa -At mempunyai diagonal nol. Maka antisymetric -At mempunyai diagonal matrik.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
A = 0 4 9 B = 0 -3 8
-4 0 5 3 0 -6
-9 -5 0 -8 6 0
(A+B) = 0 1 17
-1 0 -1
-17 1 0
(A+B)t = 0 -1 -17 -(A+B)t = 0 1 17
1 0 1 -1 0 -1
17 -1 0 -17 1 0
Dapat dilihat bahwa hasil dari (A+B) sama dengan -(A+B)t.
Maka dapat dituliskan (A+B) = – (A+B)t.
Nama : Septian Dwi Cahyono
Kelas : S1 TI 1M
NIM : 10.11.4512
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti benar)
NAMA : RIAN HIDAYAH WIHARJA
NIM : 10.11.4467
KELAS: S1 TI 1M
Pembuktian dengan menggunakan Matlab 7.10
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct antisymmetric matrix berarti benar
Nama : Rut Merilene H
NIM : 10.11.4474
Kelas : S1 TI 1M
1. Example of antisymmetric matrix:
0 -2 3
A= 2 0 -1
-2 3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
4 -3 1 0 -4 3
A= 3 5 -2 B= 4 0 -1
-1 2 3 -3 1 0
4 3 -1 0 4 -3
At= -3 3 2 Bt= -4 0 1
1-2 5 3 -1 0
-4 3 1 0 -4 3
-At=3 -3 -2 -Bt=4 0 -1
-1 2 -5 -3 1 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -2 1 0 -3 1
A= 2 0 -3 B= 3 0 -2
-1 3 0 -1 2 0
0+0 -3+(-2) 1+1
C=A+B= 2+3 0+0 -3+(-2)
-1+(-1) 3+2 0+0
0 -5 2
= 5 0 -5
-2 5 0
0 5 -2
Ct= -5 0 5
2 -5 0
0 -5 2
-Ct= 5 0 -5
-2 5 0
C = -Ct (antisymmetric matrix)
Nama : Alan Darmasaputra
NIM : 10.11.4495
Kelas : S1-TI-1M
Pembuktian dengan menggunakan Matlab 7.10
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct antisymmetric matrix berarti benar
Nama :MUHAMMAD USMAN ROSYIDI
Nim : 10.11.4489
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 4 5
A = -4 0 -8
-5 8 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 6
A = 3 1 6 B = 2 0 6
8 -6 4 -6 -6 0
7 3 -8 0 2 -4
At = -3 1 -4 Bt = -2 0 -7
8 4 4 4 7 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -9
A = 4 0 -5 , B = -7 0 8
-8 5 0 9 -8 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 6
Ct = 3 0 4
-6 -4 0
0 3 7
-Ct = -3 0 -2
-7 2 0
C = -Ct
Nama :MUHAMMAD USMAN ROSYIDI
Nim : 10.11.4489
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 4 5
A = -4 0 -8
-5 8 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 6
A = 3 1 6 B = 2 0 6
8 -6 4 -6 -6 0
7 3 -8 0 2 -4
At = -3 1 -4 Bt = -2 0 -7
8 4 4 4 7 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -9
A = 4 0 -5 , B = -7 0 8
-8 5 0 9 -8 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 6
Ct = 3 0 4
-6 -4 0
0 3 7
-Ct = -3 0 -2
-7 2 0
C = -Ct (antisymmetric matrix)
Nama :MUHAMMAD USMAN ROSYIDI
Nim : 10.11.4489
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 4 5
A = -4 0 -8
-5 8 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 6
Ct = 3 0 4
-6 -4 0
0 3 7
-Ct = -3 0 -2
-7 2 0
C = -Ct (antisymmetric matrix)
Nama :MUHAMMAD USMAN ROSYIDI
Nim : 10.11.4489
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6a
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama : M.USMAN ROSYIDI
Nim : 10.11.4489
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama :MUHAMMAD USMAN ROSYIDI
Nim : 10.11.4489
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama : MUHAMMAD USMAN ROSYIDI
Nim : 10.11.4489
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
M.Ervan Wirawan
10.11.4513
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
-8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama : pamji ginanjar setiawan
NIM : 10.11.4484
Kelas : S1 TI 1M
Pembuktian dengan menggunakan Matlab 7.10
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti bener)
Nama : panji ginanjar setiawan
Nim : 10.11.4584
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama : Panji ginanjar setiawan
NIM : 10.11.4484
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 3 -2
A = -3 0 1
2 -1 0
2. Prove that antisymmetric matrix is a square matrix with
zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
8 -4 9 0 -4 9
A = 4 5 6 B = 4 0 6
-9 -6 3 -9 -6 0
8 4 -9 0 4 -9
At = -4 5 -6 Bt = -4 0 -6
9 6 3 9 6 0
-8 -4 9 0 -4 9
-At = 4 -5 6 -Bt = 4 0 6
-9 -6 -3 -9 -6 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an
antisymmetric matrix.
0 -4 9 0 3 -8
A = 4 0 -6 , B = -3 0 7
-9 6 0 8 -7 0
0+0 -4+3 9+(-8)
C = A + B = 4+(-3) 0+0 -6+7
-9+8 6+(-7) 0+0
0 -1 1
= 1 0 1
-1 -1 0
0 1 -1
Ct = -1 0 -1
1 1 0
0 -1 1
-Ct = 1 0 1
-1 -1 0
C = -Ct (antisymmetric matrix)
Nama : panji ginanjar setiawan
NIM : 10.11.4484
Kelas : S1 TI 1M
Pembuktian dengan menggunakan Matlab 7.10
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti bener)
Nama : Faldi Dwi Hastiya
NIM : 10.11.4523
Kelas : S1 TI 1M
Pembuktian dengan menggunakan Matlab 7.10
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti bener)
Nama : Eko Whyu Nugroho
NIM : 10.11.4468
Kelas : S1 TI 1M
Pembuktian dengan menggunakan Matlab 7.10
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti bener)
Nama : Beny Viktor
NIM : 10.11.4480
Kelas : S1 TI 1M
Pembuktian dengan menggunakan Matlab 7.10
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti bener)
Nama : Dwi Prasetyo Wibowo
NIM : 10.11.4471
Kelas : S1 TI 1M
Pembuktian dengan menggunakan Matlab 7.10
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti bener)
Nama : Dwi Prosetyo Wibowo
NIM : 10.11.4471
Kelas : S1 TI 1M
Pembuktian dengan menggunakan Matlab 7.10
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti bener)
Nama : Aurelius N Taja
NIM : 10.11.4473
Kelas : 1M
Pembuktian dengan menggunakan Matlab 7.10
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti bener)
nama : panji ginanjar.s
nim : 10.11.4484
kelas : s1 ti 1m
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
NAMA : PANJI GINANJAR.S
NIM : 10.11.4484
KELAS :S1 TI 1M
1. Example of antisymmetric matrix:
0 4 -2
A = -3 2 5
1 -5 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
6 -2 1 6 5 9
A = 4 2 5 B = -2 0 -1
-8 -3 2 1 -4 2
-6 2 1 -6 -5 -9
At = -4 -2 -5 Bt = 2 0 1
8 3 -2 -1 4 -2
6 -2 1 6 5 9
-At = 4 2 5 -Bt = -2 0 -1
-8 -3 2 1 -4 2
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
1 2 1 4 -2 -5
Let A = 3 0 3 , B = -1 0 4
-2 3 0 5 -4 0
1+4 2+(-2) 1+(-5)
C = A + B = 3+(-1) 0+0 3+4
-2+5 3+(-4) 0+0
5 0 -4
= 2 0 7
3 -1 0
5 0 -4
Ct = 2 0 7
3 -1 0
-5 0 4
-Ct = -2 0 -7
-3 1 0
C = -Ct (antisymmetric matrix)
nama:panji ginanjar setiawan
NIM : 10.11.4484
kelas:s1 TI 1M
1. Example of antisymmetric matrix:
0 4 -2
A = -3 2 5
1 -5 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
6 -2 1 6 5 9
A = 4 2 5 B = -2 0 -1
-8 -3 2 1 -4 2
-6 2 1 -6 -5 -9
At = -4 -2 -5 Bt = 2 0 1
8 3 -2 -1 4 -2
6 -2 1 6 5 9
-At = 4 2 5 -Bt = -2 0 -1
-8 -3 2 1 -4 2
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
1 2 1 4 -2 -5
Let A = 3 0 3 , B = -1 0 4
-2 3 0 5 -4 0
1+4 2+(-2) 1+(-5)
C = A + B = 3+(-1) 0+0 3+4
-2+5 3+(-4) 0+0
5 0 -4
= 2 0 7
3 -1 0
5 0 -4
Ct = 2 0 7
3 -1 0
-5 0 4
-Ct = -2 0 -7
-3 1 0
C = -Ct (antisymmetric matrix)
Nama : Banter Prayogo
Nim : 10.11.4470
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama : Adnan Dwipa Hanggoro
Nim : 10.11.4539
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama : Bima Aditya Rahayu
Nim : 10.11.4476
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama : Mahasin Ahmad Alfaqh
Nim : 10.11.4530
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama : Nur Cholis
Nim : 10.11.4545
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama : Muhamad Amirudin Fauzi
NIM : 10.11.4479
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama : Muhamad Amirudin Fauzi
NIM : 10.11.4479
Kelas : S1- TI – 1M
1. Example of antisymmetric matrix:
0 3 -2
A = -3 0 1
2 -1 0
2. Prove that antisymmetric matrix is a square matrix with
zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
8 -4 9 0 -4 9
A = 4 5 6 B = 4 0 6
-9 -6 3 -9 -6 0
8 4 -9 0 4 -9
At = -4 5 -6 Bt = -4 0 -6
9 6 3 9 6 0
-8 -4 9 0 -4 9
-At = 4 -5 6 -Bt = 4 0 6
-9 -6 -3 -9 -6 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an
antisymmetric matrix.
0 -4 9 0 3 -8
A = 4 0 -6 , B = -3 0 7
-9 6 0 8 -7 0
0+0 -4+3 9+(-8)
C = A + B = 4+(-3) 0+0 -6+7
-9+8 6+(-7) 0+0
0 -1 1
= 1 0 1
-1 -1 0
0 1 -1
Ct = -1 0 -1
1 1 0
0 -1 1
-Ct = 1 0 1
-1 -1 0
C = -Ct (antisymmetric matrix)
Nama : Pemula Budi Akbar
NIM : 10.11.4536
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 3 -2
A = -3 0 1
2 -1 0
2. Prove that antisymmetric matrix is a square matrix with
zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
8 -4 9 0 -4 9
A = 4 5 6 B = 4 0 6
-9 -6 3 -9 -6 0
8 4 -9 0 4 -9
At = -4 5 -6 Bt = -4 0 -6
9 6 3 9 6 0
-8 -4 9 0 -4 9
-At = 4 -5 6 -Bt = 4 0 6
-9 -6 -3 -9 -6 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an
antisymmetric matrix.
0 -4 9 0 3 -8
A = 4 0 -6 , B = -3 0 7
-9 6 0 8 -7 0
0+0 -4+3 9+(-8)
C = A + B = 4+(-3) 0+0 -6+7
-9+8 6+(-7) 0+0
0 -1 1
= 1 0 1
-1 -1 0
0 1 -1
Ct = -1 0 -1
1 1 0
0 -1 1
-Ct = 1 0 1
-1 -1 0
C = -Ct (antisymmetric matrix)
Nama : senja permata dewanti
NIM : 10.11.4511
Kelas : S1-TI-1M
1. Example of antisymmetric matrix:
0 3 -2
A = -3 0 1
2 -1 0
2. Prove that antisymmetric matrix is a square matrix with
zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
8 -4 9 0 -4 9
A = 4 5 6 B = 4 0 6
-9 -6 3 -9 -6 0
8 4 -9 0 4 -9
At = -4 5 -6 Bt = -4 0 -6
9 6 3 9 6 0
-8 -4 9 0 -4 9
-At = 4 -5 6 -Bt = 4 0 6
-9 -6 -3 -9 -6 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an
antisymmetric matrix.
0 -4 9 0 3 -8
A = 4 0 -6 , B = -3 0 7
-9 6 0 8 -7 0
0+0 -4+3 9+(-8)
C = A + B = 4+(-3) 0+0 -6+7
-9+8 6+(-7) 0+0
0 -1 1
= 1 0 1
-1 -1 0
0 1 -1
Ct = -1 0 -1
1 1 0
0 -1 1
-Ct = 1 0 1
-1 -1 0
C = -Ct (antisymmetric matrix)
Nama : Muhyiddin Al-Idrus
NIM : 10.11.4540
Kelas : 1M
Pembuktian dengan menggunakan Matlab 7.10
1. Example of antisymmetric matrix:
>> A=[0 -2 -1;-2 0 3;1 -3 0]
A =
0 -2 -1
-2 0 3
1 -3 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements:
>> A=[6 -3 8;3 1 7;-8 -7 4],B=[0 -3 8;3 0 7;-8 -7 0]
A =
6 -3 8
3 1 7
-8 -7 4
B =
0 -3 8
3 0 7
-8 -7 0
>> A’,B’
ans =
6 3 -8
-3 1 -7
8 7 4
ans =
0 3 -8
-3 0 -7
8 7 0
>> -A’,-B’
ans =
-6 -3 8
3 -1 7
-8 -7 -4
ans =
0 -3 8
3 0 7
-8 -7 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
>> A=[0 -4 8;4 0 -5;-8 5 0],B=[0 7 -3;-7 0 6;3 -6 0]
A =
0 -4 8
4 0 -5
-8 5 0
B =
0 7 -3
-7 0 6
3 -6 0
>> C=A+B
C =
0 3 5
-3 0 1
-5 -1 0
>> C’
ans =
0 -3 -5
3 0 -1
5 1 0
>> -C’
ans =
0 3 5
-3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix berarti bener)
Nama : Muhyiddin Al-Idrus
NIM : 10.11.4540
Kelas : 1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama : Soraya Inayah Fidran
Kelas : S1 TI-M
NIM : 10.11.4522
1. Example of antisymmetric matrix:
0 4 -2
A = -3 2 5
1 -5 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
6 -2 1 6 5 9
A = 4 2 5 B = -2 0 -1
-8 -3 2 1 -4 2
-6 2 1 -6 -5 -9
At = -4 -2 -5 Bt = 2 0 1
8 3 -2 -1 4 -2
6 -2 1 6 5 9
-At = 4 2 5 -Bt = -2 0 -1
-8 -3 2 1 -4 2
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
1 2 1 4 -2 -5
Let A = 3 0 3 , B = -1 0 4
-2 3 0 5 -4 0
1+4 2+(-2) 1+(-5)
C = A + B = 3+(-1) 0+0 3+4
-2+5 3+(-4) 0+0
5 0 -4
= 2 0 7
3 -1 0
5 0 -4
Ct = 2 0 7
3 -1 0
-5 0 4
-Ct = -2 0 -7
-3 1 0
C = -Ct (antisymmetric matrix)
Nama : Muhyiddin Al-Idrus
Kelas : 1M
1. Example of antisymmetric matrix:
0 4 -3
A = -4 0 6
3 -6 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
6 -3 8 0 -2 7
A = 3 1 7 B = 2 0 6
-8 -7 4 -7 -6 0
6 3 -8 0 2 -7
At = -3 1 -7 Bt = -2 0 -6
8 7 4 7 6 0
-6 -3 8 0 -2 7
-At = 3 -1 7 -Bt = 2 0 6
-8 -7 -4 -7 -6 0
Kita bisa melihat bahwa A =-At dan B =-Bt. Jadi itu membuktikan bahwa antisimetrik matriks adalah matriks persegi dengan nol pada setiap elemen diagonal.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -4 8 0 7 -3
A = 4 0 -5 , B = -7 0 6
-8 5 0 3 -6 0
0+0 -4+7 8+(-3)
C = A + B = 4+(-7) 0+0 -5+6
-8+3 5+(-6) 0+0
0 3 5
= -3 0 1
-5 -1 0
0 -3 -5
Ct = 3 0 -1
5 1 0
0 3 5
-Ct = -3 0 1
-5 -1 0
C = -Ct (antisymmetric matrix)
Nama = Margubillah
Nim = 09.11.3475
Kelas = SITI – 3M
Visual Basik
Private Sub Command1_Click()
If Text1.Text = “bill” Then
Picture1.Visible = True
Text1.Enabled = False
Command1.Enabled = False
Else
MsgBox “password salah..”
Text1.Text = “”
End If
End Sub
==========================================
JAVA JDK
public class Proposisi {
public static void main(String args[])
{
String bill = “STMIK AMIKOM”; //Himpunan
System.out.println(“Membaca banyak himpunan berupa huruf \n================================== “);
System.out.println(“\nhimpunan x = \”” +bill +”\” size = “+bill.length()+ ” digit”);
System.out.println(“Apakah himpunan dengan kata ‘STMIK’ = ” +bill.startsWith(“STMIK”));
System.out.println(“Apakah himpunan dengan kata ‘AKAKOM’ = ” +bill.startsWith(“AKAKOM”));
System.out.println(“Apakah himpunan dengan kata ‘AMIKOM’ = ” +bill.endsWith(“AMIKOM”));
}
}
1. Example of antisymmetric matrix:
0 3 -2
A = -3 0 5
2 -5 0
2. Prove that antisymmetric matrix is a square matrix with zero on each diagonal elements.
Antisymmetric matrix is an anti of symmetric matrix, with (A= -At ). That mean the matrix is equal to its transpose, but each element of the transposed matrix has an opposite sign of the matrix.
Here are two examples of symmetric matrix, one with zero on each diagonal elements.
7 -4 9 0 -4 9
A = 4 2 8 B = 4 0 8
-9 -8 5 -9 -8 0
7 4 -9 0 4 -9
At = -4 2 -8 Bt = -4 0 -8
9 8 5 9 8 0
-7 -4 9 0 -4 9
-At = 4 -2 8 -Bt = 4 0 8
-9 -8 -5 -9 -8 0
We could see that A != -At and B = -Bt. So it’s proved that antisymmetric matrix is a square matrix with zero on each diagonal elements.
3. Prove that sum of two antisymmetric matrix is an antisymmetric matrix.
0 -3 7 0 11 -5
Let A = 3 0 -4 , B = -11 0 8
-7 4 0 5 -8 0
0+0 -3+11 7+(-5)
C = A + B = 3+(-11) 0+0 -4+8
-7+5 4+(-8) 0+0
0 8 2
= -8 0 4
-2 -4 0
0 -8 -2
Ct = 8 0 -4
2 4 0
0 8 2
-Ct = -8 0 4
-2 -4 0
C = -Ct (antisymmetric matrix) –Proved–