Aljabar matriks
250
ANALISIS STRUKTUR DENGAN METODE MATRIKS Mengapa Metode Matriks?
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Aljabar Matriks
Transcript of Aljabar matriks
ANALISIS STRUKTUR
DENGAN METODE MATRIKS
Mengapa Metode Matriks?
ALJABAR MATRIKSPertemuan Minggu I dan II
Analisis Struktur Metode Matriks
Ashar Saputra, PhD
Topik Paparan
• Aljabar matrix
• Definisi matrix
• Macam-macam matrix
• Operasi matrix
• Matrix orthogonal
• Aljabar matrix
• Solusi persamaan linier simultan
• Matrix partisi
• Program komputer untuk operasi matrix
Definisi Matriks
• Matriks adalah suatu susunan elemen dengan dobel
subscribe yang disusun dalam baris dan kolom
ij
mnm
n
n
A
aa
aa
aa
,,
,,
,,
1
221
111
A
Vektor baris
[1 x n] matrix
jn aaaaA ,, 2 ,1
Vektor kolom
i
m
a
a
a
a
A 2
1
[m x 1] matrix
Matrik bujur sangkar
B
5 4 7
3 6 1
2 1 3
Jumlah baris dan kolom sama
Macam-macam Matriks bujur sangkar
•
Macam-macam Matriks bujur sangkar
•
Macam-macam Matriks bujur sangkar
•
Macam-macam Matriks bujur sangkar
•
OPERASI MATRIKS
Penjumlahan
•
Sifat-sifat penjumlahan
• [A] + [B] = [B] + [A] komutatif
• [A] + [B] + [C] = ([A] + [B]) + [C]
PERKALIAN DENGAN SKALAR
3 2 5
-2 0 1
4 6 9
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
9
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
9 6
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
9 6 15
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
9 6 15
-6
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
9 6 15
-6 0
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
9 6 15
-6 0 3
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
9 6 15
-6 0 3
12
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
9 6 15
-6 0 3
12 18
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
9 6 15
-6 0 3
12 18 27
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
9 6 15
-6 0 3
12 18 27
Jeff Bivin -- LZHS
PENJUMLAHAN
(LANJUTAN)
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
=
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
+=
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
+ 26
=6 + 20
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
26 0
=+
4 - 4
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
26 0 18
=+
10 + 8
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
26 0 18
-4=+
-4 + 0
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
26 0 18
-4 12=+
0 + 12
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
26 0 18
-4 12 30=+
2 + 28
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
26 0 18
-4 12 30
16
=+
8 + 8
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
26 0 18
-4 12 30
16 40
=+
12 + 28
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
26 0 18
-4 12 30
16 40 34
=+
18 + 16
Jeff Bivin -- LZHS
Matrix Addition & Scalars
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
26 0 18
-4 12 30
16 40 34
=
Jeff Bivin -- LZHS
PERKALIAN MATRIKS
DENGAN MATRIKS LAIN
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
Jeff Bivin -- LZHS
Matrix Multiplication
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
Jeff Bivin -- LZHS
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
Jeff Bivin -- LZHS
CONTOH PERHITUNGAN
Jeff Bivin -- LZHS
Row 1 x Column 1
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
3 x 5 + 2 x 0 + 5 x 2 = 25
Jeff Bivin -- LZHS
Row 1 x Column 1
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
3 x 5 + 2 x 0 + 5 x 2 = 25
Jeff Bivin -- LZHS
Row 1 x Column 1
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
3 x 5 + 2 x 0 + 5 x 2 = 25
Jeff Bivin -- LZHS
Row 1 x Column 1
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25
=
Jeff Bivin -- LZHS
Row 1 x Column 2
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
3 x (-1) + 2 x 3 + 5 x 7 = 38
Jeff Bivin -- LZHS
Row 1 x Column 2
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
3 x (-1) + 2 x 3 + 5 x 7 = 38
Jeff Bivin -- LZHS
Row 1 x Column 2
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
3 x (-1) + 2 x 3 + 5 x 7 = 38
Jeff Bivin -- LZHS
Row 1 x Column 2
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38
=
Jeff Bivin -- LZHS
Row 1 x Column 3
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
3 x 2 + 2 x 7 + 5 x 4 = 40
Jeff Bivin -- LZHS
Row 1 x Column 3
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
3 x 2 + 2 x 7 + 5 x 4 = 40
Jeff Bivin -- LZHS
Row 1 x Column 3
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
3 x 2 + 2 x 7 + 5 x 4 = 40
Jeff Bivin -- LZHS
Row 1 x Column 3
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
=
Jeff Bivin -- LZHS
Row 2 x Column 1
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
-2 x 5 + 0 x 0 + 1 x 2 = -8
Jeff Bivin -- LZHS
Row 2 x Column 1
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
-2 x 5 + 0 x 0 + 1 x 2 = -8
Jeff Bivin -- LZHS
Row 2 x Column 1
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
-2 x 5 + 0 x 0 + 1 x 2 = -8
Jeff Bivin -- LZHS
Row 2 x Column 1
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
-8=
Jeff Bivin -- LZHS
Row 2 x Column 2
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
Jeff Bivin -- LZHS
Row 2 x Column 2
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
-2 x (-1) + 0 x 3 + 1 x 7 = 9
Jeff Bivin -- LZHS
Row 2 x Column 2
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
-2 x (-1) + 0 x 3 + 1 x 7 = 9
Jeff Bivin -- LZHS
Row 2 x Column 2
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
-2 x (-1) + 0 x 3 + 1 x 7 = 9
Jeff Bivin -- LZHS
Row 2 x Column 2
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
-8 9=
Jeff Bivin -- LZHS
Row 2 x Column 3
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
-2 x 2 + 0 x 7 + 1 x 4 = 0
Jeff Bivin -- LZHS
Row 2 x Column 3
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
-2 x 2 + 0 x 7 + 1 x 4 = 0
Jeff Bivin -- LZHS
Row 2 x Column 3
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
-2 x 2 + 0 x 7 + 1 x 4 = 0
Jeff Bivin -- LZHS
Row 2 x Column 3
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
-8 9 0=
Jeff Bivin -- LZHS
Row 3 x Column 1
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
4 x 5 + 6 x 0 + 9 x 2 = 38
Jeff Bivin -- LZHS
Row 3 x Column 1
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
4 x 5 + 6 x 0 + 9 x 2 = 38
Jeff Bivin -- LZHS
Row 3 x Column 1
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
4 x 5 + 6 x 0 + 9 x 2 = 38
Jeff Bivin -- LZHS
Row 3 x Column 1
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
-8 9 0
38
=
Jeff Bivin -- LZHS
Row 3 x Column 2
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
4 x (-1) + 6 x 3 + 9 x 7 = 77
Jeff Bivin -- LZHS
Row 3 x Column 2
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
4 x (-1) + 6 x 3 + 9 x 7 = 77
Jeff Bivin -- LZHS
Row 3 x Column 2
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
4 x (-1) + 6 x 3 + 9 x 7 = 77
Jeff Bivin -- LZHS
Row 3 x Column 2
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
-8 9 0
38 77
=
Jeff Bivin -- LZHS
Row 3 x Column 3
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
4 x 2 + 6 x 7 + 9 x 4 = 86
Jeff Bivin -- LZHS
Row 3 x Column 3
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
4 x 2 + 6 x 7 + 9 x 4 = 86
Jeff Bivin -- LZHS
Row 3 x Column 3
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
4 x 2 + 6 x 7 + 9 x 4 = 86
Jeff Bivin -- LZHS
Row 3 x Column 3
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
-8 9 0
38 77 86
Jeff Bivin -- LZHS
Matrix Multiplication
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
-8 9 0
38 77 86
=
Jeff Bivin -- LZHS
Matrix Multiplication
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
-8 9 0
38 77 86
=
Jeff Bivin -- LZHS
Matrix Multiplication
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
-8 9 0
38 77 86
=
Jeff Bivin -- LZHS
Review
Jeff Bivin -- LZHS
Row 1 x Column 1
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25
=
Jeff Bivin -- LZHS
Row 1 x Column 2
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38
=
Jeff Bivin -- LZHS
Row 1 x Column 3
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
=
Jeff Bivin -- LZHS
Row 2 x Column 1
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
-8=
Jeff Bivin -- LZHS
Row 2 x Column 2
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
-8 9=
Jeff Bivin -- LZHS
Row 2 x Column 3
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
-8 9 0=
Jeff Bivin -- LZHS
Row 3 x Column 1
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
-8 9 0
38
=
Jeff Bivin -- LZHS
Row 3 x Column 2
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
-8 9 0
38 77
=
Jeff Bivin -- LZHS
Row 3 x Column 3
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
-8 9 0
38 77 86
=
Jeff Bivin -- LZHS
Matrix Multiplication
3 2 5
-2 0 1
4 6 9
5 -1 2
0 3 7
2 7 4
X
25 38 40
-8 9 0
38 77 86
=
Jeff Bivin -- LZHS
Determinant dari matriks 2 x 2
a b
c d
Jeff Bivin -- LZHS
Determinant of a 2 x 2 Matrix
3 5
4 6
Jeff Bivin -- LZHS
Determinant of a 2 x 2 Matrix
-4 3
5 2
Jeff Bivin -- LZHS
Determinant of a 2 x 2 Matrix
8 4
6 5
Jeff Bivin -- LZHS
Determinant dari matriks 3 x 3
1 3 4
2 1 5
3 6 7
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1
2 1 5 2
3 6 7 3
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4 1 3
2 1 5 2 1
3 6 7 3 6
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3
7 + 45 + 48 - 12 - 30 - 42
= 16Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2
4 3 7 4
5 9 8 5
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2
4 3 7 4
5 9 8 5
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
1•1•7 + 3•5•3 + 4•2•6 - 3•1•4 - 6•5•1 - 7•2•3
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
2 1 6 2 1
4 3 7 4 3
5 9 8 5 9
2•3•8 + 1•7•5 + 6•4•9 - 5•3•6 - 9•7•2 - 8•4•1
48 + 35 + 216 - 90 - 126 - 32
= 51Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
Choose a
row or
a colum
1 3 4
2 1 5
3 6 7
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
We will use the first column
to give us our cofactors
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
1 5
6 7
3 4
6 7
3 4
1 5
Notice the
alternating signs
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
1 5
6 7
3 4
6 7
3 4
1 5
Now for the minors
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
1 5
6 7
Remove the row and the
column of the cofactor
element
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
1 5
6 7
3 4
6 7
Remove the row and the
column of the cofactor
element
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
1 5
6 7
3 4
6 7
3 4
1 5
Remove the row and the
column of the cofactor
element
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
1 5
6 7
3 4
6 7
3 4
1 51(7 – 30) – 2(21 – 24) + 3(15 –
4)1(-23) – 2(–3) + 3(11) = -23 + 6 + 33 =
16
= 16Evaluate each 2x2
determinant and simplify
Jeff Bivin -- LZHS
NOW TRY A DIFFERENT
ROW OR COLUMN
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
Choose a new
row or
a colum
1 3 4
2 1 5
3 6 7
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
This time we will use the
second row to give us our
cofactors
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
1 5
6 7
3 4
6 7
3 4
1 5
Again we have
alternating signs
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
1 5
6 7
3 4
6 7
3 4
1 5
Now for the minors
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
Remove the row and the
column of the cofactor
element
3 4
6 7
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
Remove the row and the
column of the cofactor
element
3 4
6 7
1 4
3 7
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
Remove the row and the
column of the cofactor
element
3 4
6 7
1 4
3 7
1 3
3 6
Jeff Bivin -- LZHS
Determinant of a 3 x 3 Matrix
1 3 4
2 1 5
3 6 7
3 4
6 7
1 4
3 7
1 3
3 6-2(21 – 24) + 1(7 – 12) - 5(6 –
9)-2(-3) + 1(–5) - 5(-3) = 6 - 5 + 15 = 16
= 16Evaluate each 2x2
determinant and simplify
Jeff Bivin -- LZHS
Determinant of a 4 x 4 Matrix
1 -2 3 4
2 -5 1 5
2 3 -1 4
3 1 6 7
Select a row or column
to use as the co-
factors.
Jeff Bivin -- LZHS
Determinant of a 4 x 4 Matrix
1 -2 3 4
2 -5 1 5
2 3 -1 4
3 1 6 7
Let’s use the first row for the
co-factors
Jeff Bivin -- LZHS
1 3 4-2
Determinant of a 4 x 4 Matrix
1 -2 3 4
2 -5 1 5
2 3 -1 4
3 1 6 7
= 16Remember the alternating
signs.
Jeff Bivin -- LZHS
1 -2 3 4
Determinant of a 4 x 4 Matrix
1 -2 3 4
2 -5 1 5
2 3 -1 4
3 1 6 7
-5 1 5
3 -1 4
1 6 7
Jeff Bivin -- LZHS
+ 3 - 4
Remove the row and the
column of the cofactor
element
1 + 2
Determinant of a 4 x 4 Matrix
1 -2 3 4
2 -5 1 5
2 3 -1 4
3 1 6 7
-5 1 5
3 -1 4
1 6 7
Jeff Bivin -- LZHS
1 + 22 1 5
2 -1 4
3 6 7
Remove the row and the
column of the cofactor
element
+ 3 - 4
Determinant of a 4 x 4 Matrix
1 -2 3 4
2 -5 1 5
2 3 -1 4
3 1 6 7
-5 1 5
3 -1 4
1 6 7
Jeff Bivin -- LZHS
1 + 22 -5 5
2 3 4
3 1 7
2 1 5
2 -1 4
3 6 7
Remove the row and the
column of the cofactor
element
+ 3 - 4
Determinant of a 4 x 4 Matrix
1 -2 3 4
2 -5 1 5
2 3 -1 4
3 1 6 7
-5 1 5
3 -1 4
1 6 7
Jeff Bivin -- LZHS
2 1 5
2 -1 4
3 6 7
2 -5 5
2 3 4
3 1 7
2 -5 1
2 3 -1
3 1 6
1 + 2
Remove the row and the
column of the cofactor
element
+ 3 - 4
Determinant of a 4 x 4 Matrix
1 -2 3 4
2 -5 1 5
2 3 -1 4
3 1 6 7
-5 1 5
3 -1 4
1 6 7
Jeff Bivin -- LZHS
2 1 5
2 -1 4
3 6 7
2 -5 5
2 3 4
3 1 7
2 -5 1
2 3 -1
3 1 6
1 + 2
Now evaluate the
3x3 determinants --- more
expansion by
co-factors and minors
1(233) + 2(11) + 3(9) -
4(106)
= -142
+ 3 - 4
= -142
INVERS DAN MATRIKS
IDENTITAS
Invers Matiks Bujur Sangkar
• Untuk mencari inverse matriks dapat dipakai beberapa
metoda, antara lain metode adjoint, metode pemisahan,
Gauss-Jordan, Chelosky, dsb.
• Sebagai contoh metode Gauss-Jordan
Invers Matiks Bujur Sangkar
•
Inverses and Identities
5x = 351
51
531 x
53x
inversestivemultiplicaareand 551
identitytivemultiplicatheis1
Jeff Bivin -- LZHS
10
01
Now with Matrices
43
21
43
21
This is the
Identity Matrix
for 2 x 2 Matrices
Jeff Bivin -- LZHS
Let’s look at another example
53
87
10
01
53
87
Jeff Bivin -- LZHS
New Question
43
21
10
01
What do we
multiply a matrix
by to get the
Identity?
Jeff Bivin -- LZHS
The Inverse of a 2x2 Matrix
dc
ba
dc
ba
1
ac
bd
0dc
ba
Jeff Bivin -- LZHS
The Inverse of a 2x2 Matrix
43
21
43
21
1
13
24
13
2464
1
2
1
21
23
12
Jeff Bivin -- LZHS
The Inverse of a 2x2 Matrix
21
34
21
34
1
41
32
41
3238
1
11
1
114
111
113
112
Jeff Bivin -- LZHS
:' usesLet
31
25
43
21X
BAX
BAX 1A
BAXI 1
1A
BAX 1
This is our Formula!
31
25
13
24
43
21
1X2
1
914
141821X
297
79X
Jeff Bivin -- LZHS
:' usesLet
34
27
54
31X
BXA
BXA 1A1 ABIX
1A
1 ABX
This is our Formula!
14
35
34
27
54
31
1X
1532
2343
7
1X
715
732
723
743
7
1
14
35
34
27
7
1X
Jeff Bivin -- LZHS
:' usesLet
63
14
41
32
32
51X
CBAX
)( BCAX 1A
BCAXI 1
1A
BCAX 1
This is our Formula!
41
32
63
14
12
53
32
51
1X7
1
24
42
12
5371X
68
22671X
BCAX
76
78
72
726
Jeff Bivin -- LZHS
Are the two Matrices Inverses?
85
32
25
38
10
01
Jeff Bivin -- LZHS
The product of
inverse matrices
is the identity
matrix.
Identity,
therefore,
INVERSE
Matrices
Are the two Matrices Inverses?
41
23
31
24
100
010
Jeff Bivin -- LZHS
The product of
inverse matrices
is the identity
matrix.
Not the
Identity,
therefore,
NOT
INVERSE
Matrices
46
23
Jeff Bivin -- LZHS
Does the Matrix have an Inverse?
Let’s review
the definition of
the Inverse of a
2x2 Matrix
The Inverse of a 2x2 Matrix
dc
ba
dc
ba
1
ac
bd
0dc
ba
Jeff Bivin -- LZHS
46
23Find the determinant!
46
23
Jeff Bivin -- LZHS
01212
Therefore, NO inverse!
Does the Matrix have an Inverse?
144
27Find the determinant!
144
27
Jeff Bivin -- LZHS
90898
Therefore, an inverse exists!
Does the Matrix have an Inverse?
987
654
321
Find the determinant!
Jeff Bivin -- LZHS
Therefore, NO inverse!
Does the Matrix have an Inverse?
1 2 3 1 2
4 5 6 4 5
7 8 9 7 8
1•5•9 + 2•6•7 + 3•4•8 - 7•5•3 - 8•6•1 - 9•4•2
45 + 84 + 96 - 105 - 48 - 72
243
142
231
Find the determinant!
Jeff Bivin -- LZHS
Does the Matrix have an Inverse?
1 3 2 1 3
2 4 1 2 4
3 4 2 3 4
1•4•2 + 3•1•3 + 2•2•4 - 3•4•2 - 4•1•1 - 2•2•3
8 + 9 + 16 - 24 - 4 - 12
Therefore, an inverse exists!
SOLUSI PERSAMAAN
LINEAR SIMULTAN
SOLUSI PERSAMAAN LINIER SIMULTAN
Persamaan Linier Simultan dengan n buah bilangan tak diketahui dapat dituliskan sebagai
berikut:
a11 x1 + a12 x2 + ⋯⋯+ a1n xn = b1
a11 x1 + a12 x2 + ⋯⋯+ a1n xn = b2
⋮ ⋮ ⋮ ⋮
an1 x1 + an2 x2 + ⋯⋯+ ann xn = bn
Secara matrix, persamaan-persamaan tersebut, bisa ditulis:
𝑎11
𝑎21
⋮𝑎𝑛1
𝑎12
𝑎22
⋮𝑎𝑛2
……
…
𝑎1𝑛
𝑎2𝑛
⋮𝑎𝑛𝑛
x1
x2
⋮xn
=
b1
b2
⋮bn
atau, secara lebih sederhana:
[A] {X} = {B}
[A] : matrix bujur sangkar koefisien persamaan linear
{X} : matrix kolo, dari bilangan yang tak diketahui
{B} : matrix kolom dari konstanta
SOLUSI PERSAMAAN LINIER SIMULTAN
Penyelesaian dengan metode eliminasi Gauss
Dengan cara “OPERASI BARIS”, buatlah matrix [A] menjadi “UPPER TRIANGULAR
MATRIX” (suatu matrix bujur sangkar dimana semua elemen di bawah diagonal
utama sama dengan 0)
Dengan cara eliminasi/”back substitution”, bilangan-bilangan tak diketahui dapat
diperoleh.
Contoh:
4x + 3y + z = 13
x + 2y + 3z = 14
3x + 2y + 5z = 22
4 3 11 2 33 2 5
xyz =
131422
[A] {X} = {B}
413
322
135
||| 131422
;
1
0
3
34
2
2
14
3
5
|
|
|
134
14
22
1
0
0
34
54
−14
14
114
174
|
|
|
134
434
494
;
1
0
0
34
1
−14
14
115
174
|
|
|
134
435
494
1
0
0
34
1
0
14
115
245
|
|
|
134
435
725
z = 72
5 .
5
24= 3
y = 43
5 −
11
5 . 3 = 2
x = 13
4 −
3
4 . 2 −
1
4 . 3 = 1
SOLUSI PERSAMAAN LINIER SIMULTAN
:' usesLet
11
7
54
23
y
x
BAU
BAU 1A
BAUI 1
1A
BAU 1
This is our Formula!
11
7
34
25
54
23
1U23
1
5
5723
1U
23
5
2357
Jeff Bivin -- LZHS
3x + 2y = 7
4x - 5y = 11
Solve the system
using inverse matrices
23
52357 , solution
:' usesLet
1
9
23
42
y
x
BAU
BAU 1A
BAUI 1
1A
BAU 1
This is our Formula!
1
9
23
42
23
42
1U8
1
25
1481U
825
47
Jeff Bivin -- LZHS
2x - 4y = 9
3x - 2y = 1
Solve the system
using inverse matrices
825
47 , solution
CONTOH GAUSS
Jeff Bivin -- LZHS
x + y + z = 6
4x – 8y + 4z = 12
2x – 3y + 4z = 3
1 1 1 6
4 -8 4 12
2 -3 4 3
Jeff Bivin -- LZHS
1 1 1 6
4 -8 4 12
2 -3 4 3
I am a 1.
Jeff Bivin -- LZHS
1 1 1 6
4 -8 4 12
2 -3 4 3
I need to
be 0.
I need to
be 0.
Jeff Bivin -- LZHS
1 1 1 6
0 -12 0 -12
0 -5 2 -9
4 - 4(1)
-8 - 4(1)
4 - 4(1)
12 - 4(6)
2 - 2(1)
-3 - 2(1)
4 - 2(1)
3 - 2(6)
12 4RR
13 2RR
1 1 1 6
0 -12 0 -12
0 -5 2 -9
I need to
be 1
Jeff Bivin -- LZHS
2121 R
1 1 1 6
0 1 0 1
0 -5 2 -9
1 1 1 6
0 1 0 1
0 -5 2 -9
I need to
be 0.
I need to
be 0.
Jeff Bivin -- LZHS
1 0 1 5
0 1 0 1
0 0 2 -4
1 - 0
1 - 1
1 - 0
6 - 1
0 + 5(0)
-5 + 5(1)
2 + 5(0)
-9 + 5(1)
21 RR
23 5RR
1 0 1 5
0 1 0 1
0 0 2 -4
I need to
be 1
Jeff Bivin -- LZHS
321 R
1 0 1 5
0 1 0 1
0 0 1 -2
1 0 1 5
0 1 0 1
0 0 1 -2
I need to
be 0.
I am a 0
Jeff Bivin -- LZHS
1 0 0 7
0 1 0 1
0 0 1 -2
1 - 0
0 - 0
1 - 1
5 – (-2)
31 RR
1 0 0 7
0 1 0 1
0 0 1 -2
x = 7
y = 1
z = -2
Reading the Solution
Jeff Bivin -- LZHS
Writing the Solution
x + y + z = 6
4x – 8y + 4z = 12
2x – 3y + 4z = 3
Jeff Bivin -- LZHS
EXAMPLE 2
Jeff Bivin -- LZHS
x + y + z = -2
2x - 3y + z = -11
-x + 2y - z = 8
1 1 1 -2
2 -3 1 -11
-1 2 -1 8
Jeff Bivin -- LZHS
1 1 1 -2
2 -3 1 -11
-1 2 -1 8
I am a 1.
Jeff Bivin -- LZHS
1 1 1 -2
2 -3 1 -11
-1 2 -1 8
I need to
be 0.
I need to
be 0.
Jeff Bivin -- LZHS
1 1 1 -2
0 -5 -1 -7
0 3 0 6
2 - 2(1)
-3 - 2(1)
1 - 2(1)
-11 - 2(-2)
-1 + 1
2 + 1
-1 + 1
8 + (-2)
12 2RR
13 RR
1 1 1 -2
0 -5 -1 -7
0 3 0 6
I would prefer to make the 3 a one in row
three rather than the -5 in row 2. Why?
Jeff Bivin -- LZHS
1 1 1 -2
0 3 0 6
0 -5 -1 -7
To
avoid
fractions!
We will
switch Row
2 and Row 3
1 1 1 -2
0 3 0 6
0 -5 -1 -7
I need to
be 1
Jeff Bivin -- LZHS
231 R
1 1 1 -2
0 1 0 2
0 -5 -1 -7
1 1 1 -2
0 1 0 2
0 -5 -1 -7
I need to
be 0.
I need to
be 0.
Jeff Bivin -- LZHS
1 0 1 -4
0 1 0 2
0 0 -1 3
1 - 0
1 - 1
1 - 0
-2 - 2
0 + 5(0)
-5 + 5(1)
-1 + 5(0)
-7 + 5(2)
21 RR
23 5RR
1 0 1 -4
0 1 0 2
0 0 -1 3
I need to
be 1
Jeff Bivin -- LZHS
31R
1 0 1 -4
0 1 0 2
0 0 1 -3
1 0 1 -4
0 1 0 2
0 0 1 -3
I need to
be 0.
I am a 0
Jeff Bivin -- LZHS
1 0 0 -1
0 1 0 2
0 0 1 -3
1 - 0
0 - 0
1 - 1
-4 – (-3)
31 RR
1 0 0 -1
0 1 0 2
0 0 1 -3
x = -1
y = 2
z = -3
Reading the Solution
Jeff Bivin -- LZHS
Writing the Solution
Jeff Bivin -- LZHS
x + y + z = -2
2x - 3y + z = -11
-x + 2y - z = 8
AX + BY = C
DX + EY = F
43
51
49
26
84
31YX
73
21
30
19
23
75YX
AX + BY = C DX + EY = F
AX = C - BY DX = F - EY
X = A-1(C – BY) X = D-1(F – EY)
A-1(C – BY) = D-1(F – EY)
A-1C – A-1BY = D-1F – D-1EY
D-1EY – A-1BY = D-1F – A-1C
(D-1E – A-1B)Y = (D-1F – A-1C)
Y = (D-1E – A-1B)-1(D-1F – A-1C)
X = X
AX + BY = C
BY = C - AX EY = F - DX
Y = B-1(C – AX) Y = E-1(F – DX)
B-1(C – AX) = E-1(F – DX)
B-1C – B-1AX = E-1F – E-1DX
E-1DX – B-1AX = E-1F – B-1C
(E-1D – B-1A)X = (E-1F – B-1C)
X = (E-1D – B-1A)-1(E-1F – B-1C)
DX + EY = F
Y = Y
43
51
49
26
84
31YX
73
21
30
19
23
75YX
Y = (D-1E – A-1B)-1(D-1F – A-1C)
X = (E-1D – B-1A)-1(E-1F – B-1C)
73359
739
73210
7350
X
73141
7329
219392
21977
Y
AX + BY = C DX + EY = F
AX = C - BY
X = A-1(C – BY)
D[ A-1(C – BY) ] + EY= F
D[A-1C – A-1BY] + EY = F
DA-1C – DA-1BY + EY = F
EY – DA-1BY = F - DA-1C
Y = (E – DA-1B)-1(F - DA-1C)
(E – DA-1B)Y = F - DA-1C
AX + BY = C
BY = C - AX
Y = B-1(C – AX)
DX + E [ B-1(C - AX ] = F
DX + E [ B-1C - B-1AX ] = F
DX + EB-1C - EB-1AX = F
DX - EB-1AX = F - EB-1C
X = (D - EB-1A)-1(F - EB-1C)
DX + EY = F
(D - EB-1A)X = F - EB-1C
43
51
49
26
84
31YX
73
21
30
19
23
75YX
X = (D - EB-1A)-1(F - EB-1C)
Y = (E – DA-1B)-1(F - DA-1C)
73359
739
73210
7350
X
73141
7329
219392
21977
Y
ALJABAR MATRIKS
(LANJUTAN)Ashar Saputra, PhD
Transpose Matriks
•
Matriks Ortogonal
•
Teori Dekomposisi MatriksBila : [A] = sebuah matrix bujur sangkar =
𝑎11 𝑎21 𝑎31
⋮𝑎𝑛1
𝑎12 𝑎22
𝑎32 ⋮
𝑎𝑛2
𝑎13
𝑎23
𝑎33
⋮ 𝑎𝑛3
… … …
…
𝑎1𝑛
𝑎2𝑛
𝑎3𝑛
⋮ 𝑎𝑛𝑛
nxn
maka matrix tersebut dapat diekspresikan dalam bentuk:
[A] = [L][U]
dimana:
[L] =
𝑙11
𝑙21
𝑙31
⋮
𝑙𝑛1
0
𝑙22
𝑙32
⋮
𝑙𝑛2
0
0
𝑙33
⋮
𝑙𝑛3
…
…
…
…
0
0
0
⋮
𝑙𝑛𝑛
=
[L] =
𝑢11
0
0
⋮
0
𝑢12
𝑢22
0
⋮
0
𝑢13
𝑢23
𝑢33
⋮
0
…
…
…
…
𝑢1𝑛
𝑢2𝑛
𝑢3𝑛
⋮
𝑢𝑛𝑛
=
“lower triangle matrix”, matrix bujur sangkar yang semua elemen di atas/di kanan diagonal utama = 0
“upper triangle matrix”, matrix bujur sangkar yang semua elemen di bawah/di kiri diagonal utama = 0
Teori Dekomposisi Matriks
Aplikasi pada solusi persamaan linier simultan
[A] {X} = {B}
[L] [U] {X} = {B} [U]{X} = {Y}
[L] {Y} = {B} {X} = [U]-1 . {Y}
{Y} = [L]-1 . {B} {X} = [U]-1 . ([L]-1 {B})
Dengan cara ini, {X} bisa diperoleh dengan cepat/mudah tanpa harus menghitung
inverse matrix [A]
(menghemat memori dan running komputer)
Dipakai pada:
Metode eleminasi Gauss
Metode Cholesky
dapat diperoleh tanpa INVERSE
dapat diperoleh tanpa INVERSE
Teori Dekomposisi Matriks
Teori Dekomposisi Matriks
Pada analisis struktur dengan metode matrix, akan selalu dijumpai matrix (kekakuan)
yang simetris.
Bila [A] = matrix bujur sangkar dan simetris, maka:
[A] = [L] [L]T ..... ()
𝑎11
𝑎21
⋮
𝑎𝑛1
𝑎12
𝑎22
⋮
𝑎𝑛2
𝑎13
𝑎23
⋮
𝑎𝑛3
…
…
…
𝑎1𝑛
𝑎2𝑛
⋮
𝑎𝑛𝑛
=
𝑙11
𝑙21
⋮
𝑙𝑛1
0
𝑙22
⋮
𝑙𝑛2
0
0
⋮
𝑙𝑛3
…
…
…
0
0
⋮
𝑙𝑛𝑛
𝑙11
0
⋮
0
𝑙12
𝑙22
⋮
0
𝑙13
𝑙23
⋮
0
…
…
…
𝑙1𝑛
𝑙2𝑛
⋮
𝑙𝑛𝑛
𝑎11
𝑎21
⋮
𝑎𝑛1
𝑎12
𝑎22
⋮
𝑎𝑛2
𝑎13
𝑎23
⋮
𝑎𝑛3
…
…
…
𝑎1𝑛
𝑎2𝑛
⋮
𝑎𝑛𝑛
=
𝑙11
𝑙21
⋮
𝑙𝑛1
0
𝑙22
⋮
𝑙𝑛2
0
0
⋮
𝑙𝑛3
…
…
…
0
0
⋮
𝑙𝑛𝑛
𝑙11
0
⋮
0
𝑙12
𝑙22
⋮
0
𝑙13
𝑙23
⋮
0
…
…
…
𝑙1𝑛
𝑙2𝑛
⋮
𝑙𝑛𝑛
sehingga:
𝑎11 = 𝑙112 𝑙11 = 𝑎11
𝑎21 = 𝑙11 . 𝑙21 𝑙21 = 𝑎21
𝑙11
⋮ ⋮
𝑎𝑛1 = 𝑙11 . 𝑙𝑛1 𝑙𝑛1 = 𝑎𝑛1
𝑙11
𝑎22 = 𝑙21 . 𝑙21 + 𝑙22 . 𝑙22 𝑙21 = (𝑎21 − 𝑙212)
1
2
𝑎32 = 𝑙21 . 𝑙31 + 𝑙22 . 𝑙32 𝑙32 =𝑎32− 𝑙21 . 𝑙31
𝑙22
⋮ ⋮
𝑎𝑛2 = 𝑙21 . 𝑙𝑛1 + 𝑙22 . 𝑙𝑛2 𝑙𝑛2 =𝑎𝑛2− 𝑙21 . 𝑙𝑛1
𝑙22
dan seterusnya: 𝑙𝑛3 =𝑎𝑛3− 𝑙31 . 𝑙𝑛1− 𝑙32 . 𝑙𝑛2
𝑙33
Secara umum:
𝑙𝑖𝑖 = 𝑎𝑖𝑖 − . 𝑙𝑖𝑟2𝑖−1
𝑟=1 1
2
𝑙𝑖𝑗 =𝑎𝑖𝑗− . 𝑙𝑖𝑟 . 𝑙𝑗𝑟
𝑗−1𝑟=1
𝑙𝑗𝑗 untuk i > j
𝑙𝑖𝑗 = 0 untuk i < j
Pada analisis struktur dengan metode matrix, akan selalu dijumpai matrix (kekakuan)
yang simetris.
Bila [A] = matrix bujur sangkar dan simetris, maka:
[A] = [L] [L]T ..... ()
Persamaan () dapat ditulis: [A]−1 = L L T −1
= L T −1
. L −1
= L −1 T . L −1
Bila: [M] = [L]−1
Maka: [A]−1 = [M]T . M dan L [L]−1 = [I] () [L] [M] = [I]
𝑙11
𝑙21
𝑙31
⋮
𝑙𝑛1
0
𝑙22
𝑙32
⋮
𝑙𝑛2
0
0
𝑙33
⋮
𝑙𝑛3
…
…
…
…
0
0
0
⋮
𝑙𝑛𝑛
𝑚11
𝑚21
𝑚31
⋮
𝑚𝑛1
0
𝑚22
𝑚32
⋮
𝑚𝑛2
0
0
𝑚33
⋮
𝑚𝑛3
…
…
…
…
0
0
0
⋮
𝑚𝑛𝑛
=
1
0
0
⋮
0
0
1
0
⋮
0
0
0
1
⋮
0
…
…
…
…
0
0
0
⋮
1
Sehingga:
𝑙11 .𝑚11 = 1 𝑚11 = 1
𝑙11
𝑙21 .𝑚11 + 𝑙22 .𝑚22 = 0 𝑚21 = − 𝑙21 . 𝑚11
𝑙21
𝑙31 .𝑚11 + 𝑙32 .𝑚21 + 𝑙33 .𝑚31 = 0 𝑚31 = − 𝑙31 . 𝑚11 + 𝑙32 . 𝑚21
𝑙33
⋮ ⋮
Secara umum:
𝑚𝑖𝑖 = 1
𝑙𝑖𝑖
𝑚𝑖𝑗 = . 𝑙𝑖𝑟 . 𝑚𝑟𝑗𝑖−1𝑟=1
𝑙𝑖𝑖 untuk i > j
𝑚𝑖𝑗 = 0 untuk i < j
Setelah diperoleh matrix [M], maka dengan persamaan (), inverse matrix [A] dapat
dihitung:
[A]−1 = [M]T . M Metode Cholesky
PARTIONING OF MATRICES
Suatu matrix bisa dipartisikan menjadi SUB MATRIX, dengan cara mengikutkan
hanya beberapa baris atau kolom dari matrix aslinya.
Masing-masing garis partisi harus memotong suatu baris/kolom dari matrix
aslinya.
Contoh:
[A] =
𝑎11
𝑎21
𝑎31
𝑎12
𝑎22
𝑎32
𝑎13
𝑎23
𝑎33
𝑎14
𝑎24
𝑎34
𝑎15
𝑎25
𝑎35
𝑎16
𝑎26
𝑎36
= 𝐴11
𝐴21
𝐴12
𝐴22
𝐴13
𝐴23
dimana:
𝐴11 = 𝑎11
𝑎21
; 𝐴12 = 𝑎12
𝑎22
𝑎13
𝑎23
𝑎14
𝑎24
𝐴21 = 𝑎31 ; 𝐴22 = 𝑎32 𝑎33 𝑎34
....dst!
Aturan-aturan yang dipakai untuk mengoperasikan matrix partisi persis sama
dengan mengoperasikan matrix biasa.
Contoh:
𝐴 = 5 3 14 6 2
10 3 4
3𝑥3
; 𝐵 = 1 52 43 2
3𝑥2
= 𝐵1
𝐵2
2𝑥1
= 𝐴11 𝐴12
𝐴21 𝐴22
2𝑥2
𝐴 𝐵 = 𝐴11 𝐴12
𝐴21 𝐴22
𝐵1
𝐵2
= 𝐴11 .𝐵1 + 𝐴12 .𝐵2
𝐴21 .𝐵1 + 𝐴22 .𝐵2
𝐴11 𝐵1 = 5 34 6
1 52 4
= 11 3716 44
A12 B2 = 12 3 2 =
3 26 4
A21 B1 = 10 3 1 52 4
= 16 62
A22 B2 = 4 3 2 = 12 8
𝐴 𝐵 = 14 3922 4828 70
Matrix Operations in Excel
Select the
cells in
which the
answer
will
appear
Matrix Multiplication in Excel
1) Enter
“=mmult(“
2) Select the
cells of the
first matrix
3) Enter comma
“,”
4) Select the
cells of the
second matrix
5) Enter “)”
Matrix Multiplication in Excel
Enter these
three
key
strokes
at the
same
time:
control
shift
enter
Matrix Inversion in Excel
• Follow the same procedure
• Select cells in which answer is to be displayed
• Enter the formula: =minverse(
• Select the cells containing the matrix to be inverted
• Close parenthesis – type “)”
• Press three keys: Control, shift, enter
