ANALISIS STRUKTUR DENGAN METODE MATRIKS Mengapa Metode Matriks?
AL A LJABAR MATRIKS Pertemuan Minggu 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
a1 1 ,, a1n a 2 1 , , a 2 n Aij A am1 ,, amn
Vekt ektor or ba bari ris s
[1 x n] matrix
A a1, a 2 ,, an a j
Vektor kolom [m x 1] matrix
a1 a 2 A ai am
Matrik bujur sangkar
Jumlah baris dan kolom sama
B
5 3 2
4 7 6 1 1
3
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
3
2
5
-2
0
1
4
6
9
9
3
2
5
-2
0
1
4
6
9
9
6
3
2
5
-2
0
1
4
6
9
9
6
15
3
2
5
9
-2
0
1
-6
4
6
9
6
15
3
2
5
9
6
-2
0
1
-6
0
4
6
9
15
3
2
5
9
6
15
-2
0
1
-6
0
3
4
6
9
3
2
5
9
6
15
-2
0
1
-6
0
3
4
6
9
12
3
2
5
9
6
15
-2
0
1
-6
0
3
4
6
9
12
18
3
2
5
9
6
15
-2
0
1
-6
0
3
4
6
9
12
18
27
3
2
5
9
6
15
-2
0
1
-6
0
3
4
6
9
12
18
27
PENJUMLAHAN (LANJUTAN)
3
2
5
5
-1
2
-2
0
1
0
3
7
4
6
9
2
7
4
=
3
2
5
5
-1
2
-2
0
1
0
3
7
4
6
9
2
7
4
+ =
3
2
5
5
-1
2
-2
0
1
0
3
7
4
6
9
2
7
4
+ 6 + 20
26
=
3
2
5
5
-1
2
-2
0
1
0
3
7
4
6
9
2
7
4
+ 4-4
26
=
0
3
2
5
5
-1
2
-2
0
1
0
3
7
4
6
9
2
7
4
+ 10 + 8
26
=
0
18
3
2
5
5
-1
2
-2
0
1
0
3
7
4
6
9
2
7
4
+ -4 + 0
26
=
-4
0
18
3
2
5
5
-1
2
-2
0
1
0
3
7
4
6
9
2
7
4
+ 0 + 12
=
26
0
-4
12
18
3
2
5
5
-1
2
-2
0
1
0
3
7
4
6
9
2
7
4
+ 2 + 28
=
26
0
18
-4
12
30
3
2
5
5
-1
2
-2
0
1
0
3
7
4
6
9
2
7
4
+ 8+8
=
26
0
18
-4
12
30
16
3
2
5
5
-1
2
-2
0
1
0
3
7
4
6
9
2
7
4
+ 12 + 28
=
26
0
18
-4
12
30
16
40
3
2
5
5
-1
2
-2
0
1
0
3
7
4
6
9
2
7
4
+ 18 + 16
=
26
0
18
-4
12
30
16
40
34
Matrix Addition & Scalars 3
2
5
5
-1
2
-2
0
1
0
3
7
4
6
9
2
7
4
=
26
0
18
-4
12
30
16
40
34
PERKALIAN MATRIKS DENGAN MATRIKS LAIN
3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
Matrix Multiplication 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
CONTOH PERHITUNGAN
Row 1 x Column 1 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
3 x 5 + 2 x 0 + 5 x 2 = 25
Row 1 x Column 1 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
3 x 5 + 2 x 0 + 5 x 2 = 25
Row 1 x Column 1 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
3 x 5 + 2 x 0 + 5 x 2 = 25
Row 1 x Column 1 3
2
5
-2
0
1
4
6
9
X 25
=
5
-1
2
0
3
7
2
7
4
Row 1 x Column 2 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
3 x (-1) + 2 x 3 + 5 x 7 = 38
Row 1 x Column 2 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
3 x (-1) + 2 x 3 + 5 x 7 = 38
Row 1 x Column 2 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
3 x (-1) + 2 x 3 + 5 x 7 = 38
Row 1 x Column 2 3
2
5
-2
0
1
4
6
9
X 25
=
38
5
-1
2
0
3
7
2
7
4
Row 1 x Column 3 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
3 x 2 + 2 x 7 + 5 x 4 = 40
Row 1 x Column 3 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
3 x 2 + 2 x 7 + 5 x 4 = 40
Row 1 x Column 3 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
3 x 2 + 2 x 7 + 5 x 4 = 40
Row 1 x Column 3 3
2
5
-2
0
1
4
6
9
X 25
=
38
5
-1
2
0
3
7
2
7
4
40
Row 2 x Column 1 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
-2 x 5 + 0 x 0 + 1 x 2 = -8
Row 2 x Column 1 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
-2 x 5 + 0 x 0 + 1 x 2 = -8
Row 2 x Column 1 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
-2 x 5 + 0 x 0 + 1 x 2 = -8
Row 2 x Column 1 3
2
5
-2
0
1
4
6
9
X 25
=
-8
38
5
-1
2
0
3
7
2
7
4
40
Row 2 x Column 2 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
Row 2 x Column 2 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
-2 x (-1) + 0 x 3 + 1 x 7 = 9
Row 2 x Column 2 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
-2 x (-1) + 0 x 3 + 1 x 7 = 9
Row 2 x Column 2 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
-2 x (-1) + 0 x 3 + 1 x 7 = 9
Row 2 x Column 2 3
2
5
-2
0
1
4
6
9
=
X 25
38
-8
9
5
-1
2
0
3
7
2
7
4
40
Row 2 x Column 3 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
-2 x 2 + 0 x 7 + 1 x 4 = 0
Row 2 x Column 3 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
-2 x 2 + 0 x 7 + 1 x 4 = 0
Row 2 x Column 3 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
-2 x 2 + 0 x 7 + 1 x 4 = 0
Row 2 x Column 3 3
2
5
-2
0
1
4
6
9
=
X
5
-1
2
0
3
7
2
7
4
25
38
40
-8
9
0
Row 3 x Column 1 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
4 x 5 + 6 x 0 + 9 x 2 = 38
Row 3 x Column 1 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
4 x 5 + 6 x 0 + 9 x 2 = 38
Row 3 x Column 1 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
4 x 5 + 6 x 0 + 9 x 2 = 38
Row 3 x Column 1 3
2
5
-2
0
1
4
6
9
=
X
5
-1
2
0
3
7
2
7
4
25
38
40
-8
9
0
38
Row 3 x Column 2 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
4 x (-1) + 6 x 3 + 9 x 7 = 77
Row 3 x Column 2 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
4 x (-1) + 6 x 3 + 9 x 7 = 77
Row 3 x Column 2 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
4 x (-1) + 6 x 3 + 9 x 7 = 77
Row 3 x Column 2 3
2
5
-2
0
1
4
6
9
=
X
5
-1
2
0
3
7
2
7
4
25
38
40
-8
9
0
38
77
Row 3 x Column 3 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
4 x 2 + 6 x 7 + 9 x 4 = 86
Row 3 x Column 3 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
4 x 2 + 6 x 7 + 9 x 4 = 86
Row 3 x Column 3 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
4 x 2 + 6 x 7 + 9 x 4 = 86
Row 3 x Column 3 3
2
5
-2
0
1
4
6
9
X
5
-1
2
0
3
7
2
7
4
25
38
40
-8
9
0
38
77
86
Matrix Multiplication 3
2
5
-2
0
1
4
6
9
=
X
5
-1
2
0
3
7
2
7
4
25
38
40
-8
9
0
38
77
86
Matrix Multiplication 3
2
5
-2
0
1
4
6
9
=
X
5
-1
2
0
3
7
2
7
4
25
38
40
-8
9
0
38
77
86
Matrix Multiplication 3
2
5
-2
0
1
4
6
9
=
X
5
-1
2
0
3
7
2
7
4
25
38
40
-8
9
0
38
77
86
Review
Row 1 x Column 1 3
2
5
-2
0
1
4
6
9
X 25
=
5
-1
2
0
3
7
2
7
4
Row 1 x Column 2 3
2
5
-2
0
1
4
6
9
X 25
=
38
5
-1
2
0
3
7
2
7
4
Row 1 x Column 3 3
2
5
-2
0
1
4
6
9
X 25
=
38
5
-1
2
0
3
7
2
7
4
40
Row 2 x Column 1 3
2
5
-2
0
1
4
6
9
X 25
=
-8
38
5
-1
2
0
3
7
2
7
4
40
Row 2 x Column 2 3
2
5
-2
0
1
4
6
9
=
X 25
38
-8
9
5
-1
2
0
3
7
2
7
4
40
Row 2 x Column 3 3
2
5
-2
0
1
4
6
9
=
X
5
-1
2
0
3
7
2
7
4
25
38
40
-8
9
0
Row 3 x Column 1 3
2
5
-2
0
1
4
6
9
=
X
5
-1
2
0
3
7
2
7
4
25
38
40
-8
9
0
38
Row 3 x Column 2 3
2
5
-2
0
1
4
6
9
=
X
5
-1
2
0
3
7
2
7
4
25
38
40
-8
9
0
38
77
Row 3 x Column 3 3
2
5
-2
0
1
4
6
9
=
X
5
-1
2
0
3
7
2
7
4
25
38
40
-8
9
0
38
77
86
Matrix Multiplication 3
2
5
-2
0
1
4
6
9
=
X
5
-1
2
0
3
7
2
7
4
25
38
40
-8
9
0
38
77
86
Determinant dari matriks 2 x 2 a
b
c
d
Determinant of a 2 x 2 Matrix 3
5
4
6
Determinant of a 2 x 2 Matrix -4
3
5
2
Determinant of a 2 x 2 Matrix 8
4
6
5
Determinant dari matriks 3 x 3 1
3
4
2
1
5
3
6
7
Determinant of a 3 x 3 Matrix 1
3
4
2
1
5
3
6
7
Determinant of a 3 x 3 Matrix 1
3
4
1
2
1
5
2
3
6
7
3
Determinant of a 3 x 3 Matrix 1
3
4
1
3
2
1
5
2
1
3
6
7
3
6
Determinant of a 3 x 3 Matrix 1
3
4
1
3
2
1
5
2
1
3
6
7
3
6
Determinant of a 3 x 3 Matrix 1
3
4
1
3
2
1
5
2
1
3
6
7
3
6
Determinant of a 3 x 3 Matrix 1
3
4
1
3
2
1
5
2
1
3
6
7
3
6
Determinant of a 3 x 3 Matrix 1
3
4
1
3
2
1
5
2
1
3
6
7
3
6
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
= 16
Determinant of a 3 x 3 Matrix 2
1
6
2
4
3
7
4
5
9
8
5
Determinant of a 3 x 3 Matrix 2
1
6
2
4
3
7
4
5
9
8
5
Determinant of a 3 x 3 Matrix 2
1
6
2
1
4
3
7
4
3
5
9
8
5
9
Determinant of a 3 x 3 Matrix 2
1
6
2
1
4
3
7
4
3
5
9
8
5
9
Determinant of a 3 x 3 Matrix 2
1
6
2
1
4
3
7
4
3
5
9
8
5
9
Determinant of a 3 x 3 Matrix 2
1
6
2
1
4
3
7
4
3
5
9
8
5
9
Determinant of a 3 x 3 Matrix 2
1
6
2
1
4
3
7
4
3
5
9
8
5
9
Determinant of a 3 x 3 Matrix 2
1
6
2
1
4
3
7
4
3
5
9
8
5
9
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
= 51
Determinant of a 3 x 3 Matrix 1
3
4
2
1
5
3
6
7
Choose a row or a colum
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
Determinant of a 3 x 3 Matrix 1
3
4
2
1
5
3
6
7
Notice the alternating signs
1
5
3
4
3
4
6
7
6
7
1
5
Determinant of a 3 x 3 Matrix 1
3
4
2
1
5
3
6
7
Now for the minors
1
5
3
4
3
4
6
7
6
7
1
5
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
1
5
6
7
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
1
5
3
4
6
7
6
7
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
1
5
3
4
3
4
6
7
6
7
1
5
Determinant of a 3 x 3 Matrix 1
3
4
2
1
5
3
6
7
Evaluate each 2x2 determinant and simplify
= 16
1
5
3
4
3
4
6
7
6
7
1
5
1(7 – 30) – 2(21 – 24) + 3(15 – 1(-23) – 4) 2( –3) + 3(11) = -23 + 6 + 33 =
NOW TRY A DIFFERENT ROW OR COLUMN
Determinant of a 3 x 3 Matrix 1
3
4
2
1
5
3
6
7
Choose a new row or a colum
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
Determinant of a 3 x 3 Matrix 1
3
4
2
1
5
3
6
7
Again we have alternating signs
1
5
3
4
3
4
6
7
6
7
1
5
Determinant of a 3 x 3 Matrix 1
3
4
2
1
5
3
6
7
Now for the minors
1
5
3
4
3
4
6
7
6
7
1
5
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
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
1
4
6
7
3
7
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
1
4
1
3
6
7
3
7
3
6
Determinant of a 3 x 3 Matrix 1
3
4
2
1
5
3
6
7
Evaluate each 2x2 determinant and simplify
= 16
3
4
1
4
1
3
6
7
3
7
3
6
-2(21 – 24) + 1(7 – 12) - 5(6 – -2(-3) 9) + 1( –5) - 5(-3) = 6 - 5 + 15 = 16
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 cofactors.
Determinant of a 4 x 4 Matrix 1
1
-2
3
4
2
-5
1
5
2
3
-1
4
3
1
6
7
-2
Let’s use the first row for the co-factors
3
4
Determinant of a 4 x 4 Matrix 1
1
-2
3
4
2
-5
1
5
2
3
-1
4
3
1
6
7
-2
Remember the alternating signs.
= 16 3
4
Determinant of a 4 x 4 Matrix 1
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
+2
Remove the row and the column of the cofactor element
+3
-4
Determinant of a 4 x 4 Matrix 1
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
+2
Remove the row and the column of the cofactor element
2
1
5
2
-1
4
3
6
7
+3
-4
Determinant of a 4 x 4 Matrix 1
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
+2
Remove the row and the column of the cofactor element
2
1
5
2
-1
4
3
6
7
+3
2
-5
5
2
3
4
3
1
7
-4
Determinant of a 4 x 4 Matrix 1
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
+2
Remove the row and the column of the cofactor element
2
1
5
2
-1
4
3
6
7
+3
2
-5
5
2
3
4
3
1
7
-4
2
-5
1
2
3
-1
3
1
6
Determinant of a 4 x 4 Matrix 1
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
+2
Now evaluate the 3x3 determinants --- more expansion by co-factors and minors
= -142
2
1
5
2
-1
4
3
6
7
+3
2
-5
5
2
3
4
3
1
7
1(233) + 2(11) + 3(9) -
-4
2
-5
1
2
3
-1
3
1
6
= -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 1 5
5x = 3
1 x
x 1 5
1 5
3 5 3 5
and 5 are multiplicative inverses 1 is the multiplicative identity
Now with Matrices
1 0 1 2 0 1 3 4
1 2 3 4
This is the Identity Matrix for 2 x 2 Matrices
Let’s look at another example
7 8 3 5
New Question
1 0 1 2 0 1 3 4 What do we multiply a matrix by to get the Identity?
The Inverse of a 2x2 Matrix
a b c d
1
a
b
c
d
a
b
c
d
0
d b c a
The Inverse of a 2x2 Matrix
1 2 3 4 1 1 4 2 3 1 4 2 6
1 1
2
3
4
4 2 3 1
2 3 2
1
2
1
The Inverse of a 2x2 Matrix
4 3 1 2 1 1
3 811
1
2 3 1 4
4
3
1
2
2 3 1 4
1 11 2 11
4 11 3 11
X
11
1 22 3
4
4 2 5 2 3 1 1 3
18 14 X 2 14 9 1
This is our Formula!
9 X 7
7
2
9
1 3 7 2 X 4 5 4 3 7 2 X 4 3
Let ' s use : XA XA A
1
X I X
B
1 1 1 3 4
75
5 3 4 1
1
B A
B A
B A
1 1
This is our Formula! X
1 43
7 32
15 23
7 32 7 43
15 7 23 7
X
AX C B
1 1
175 2 3
3 5 4 1 2 3 2 1 3 6 1 4
3 5 2 1 X 7 2 1 4
26 1 X 7 8
This is our Formula!
2
6
4
2
7 8 7
26
6 7 2
7
Are the two Matrices Inverses?
2 3 8 3 1 0 5 8 5 2 0 1 The product of inverse matrices is the identity matrix.
Identity, therefore, INVERSE Matrices
Are the two Matrices Inverses?
3 2 4 2 10 0 1 4 1 3 0 10 The product of inverse matrices is the identity matrix.
Not the Identity, therefore, NOT INVERSE Matrices
Does the Matrix have an Inverse?
3 2 6 4
Let’s review the definition of the Inverse of a 2x2 Matrix
The Inverse of a 2x2 Matrix
a b c d
1
a
b
c
d
a
b
c
d
0
d b c a
Does the Matrix have an Inverse?
3 2 6 4
Find the determinant!
3
2
6
4
12
12 0
Therefore, NO inverse!
Does the Matrix have an Inverse?
7 2 4 14
Find the determinant!
7
2
4
14
98
8 90
Therefore, an inverse exists!
Does the Matrix have an Inverse? 1 2 3 4 5 6 7 8 9
1 2 3 1 2 4 5 6 4 5 7 8 9 7 8
Find the determinant!
1•5•9 + 2•6•7 + 3•4•8 - 7•5•3 - 8•6•1 - 9•4•2 45 + 84 + 96 - 105 - 48 - 72
Therefore, NO inverse!
Does the Matrix have an Inverse? 1 3 2 2 4 1 3 4 2
1 3 2 1 3 2 4 1 2 4 3 4 2 3 4
Find the determinant!
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:
⋮ ⋮ 11
12
21
22
1
2
1
2
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 1 3
3 2 2
1 3 5
x 13 y = 14 z 22
[A] {X} = {B}
SOLUSI PERSAMAAN LINIER SIMULTAN
1
4 3 1 | 13 1 2 3 | 14 3 2 5 | 22
− 1
3
0
5
1
3
1
0
0
z = y = x =
|
13
4
|
43
4
|
49
17
4
1
4
0
4
11
4
11 24
72 5 43 5 13 4
1
4
4
|
13
4
0
2
3
|
14
3
2
5
|
22
− 1
4
1
0
;
3
.
4
5 5
5 24
|
13
|
43
|
72
11 5
3 4
4 4
4 5 5
=3
− − −
.3=2
.2
1 4
1
4
.3=1
;
3
4
0
1
0
1
1
11
4
17
4
5 4
|
13
|
43
|
49
4 5 4
Solve the system using inverse matrices 3x + 2y = 7 4x - 5y = 11
5 2 7 U 323 2 4 3 11 4 5
1
1
57 23 1 U 23 5 5 23 57
This is our Formula!
solution
57 23
, 235
Solve the system using inverse matrices 2x - 4y = 9 3x - 2y = 1 U
1 1
2 8 4 3
2
2 3
4
9 2 1 7
14 4 1 25 U 8 25 8 This is our Formula!
solution
7 4
, 825
CONTOH GAUSS
x + y + z=6 4x – 8y + 4z = 12 2x – 3y + 4z = 3
1
1
1
6
4
-8
4
12
2
-3
4
3
I am a 1.
1
1
1
6
4
-8
4
12
2
-3
4
3
I need to be 0.
I need to be 0.
12 - 4(6) 4 - 4(1) -8 - 4(1) 4 - 4(1)
1
1
1
6
4
-8
4
12
R2 4 R1
2
-3
4
3
R3 2 R1
1
1
1
6
2 - 2(1) -3 - 2(1) 4 - 2(1)
3 - 2(6)
0
-12
0
-12
0
-5
2
-9
I need to be 1
1
1
1
6
0
-12
0
-12
0
-5
2
-9
1
1
1
6
0
1
0
1
0
-5
2
-9
1 12
R2
I need to be 0.
I need to be 0.
1
1
1
6
0
1
0
1
0
-5
2
-9
1
0
1
5
0
1
0
1
0
0
2
-4
R1 R2
R3 5 R2 0 + 5(0) -5 + 5(1) 2 + 5(0)
-9 + 5(1)
6-1 1-0 1-1 1-0
I need to be 1
1
0
1
5
0
1
0
1
0
0
2
-4
1
0
1
5
0
1
0
1
0
0
1
-2
1 2
R3
I am a 0
I need to be 0.
1
0
1
5
0
1
0
1
0
0
1
-2
1
0
0
7
0
1
0
1
0
0
1
-2
R1 R3 5 – (-2) 1-1 0-0 1-0
Reading the Solution 1
0
0
7
x=7
0
1
0
1
y=1
0
0
1
-2
z = -2
Writing the Solution x + y + z = 6 4x – 8y + 4z = 12 2x – 3y + 4z = 3
EXAMPLE 2
x + y + z = -2 2x - 3y + z = -11 -x + 2y - z = 8
1
1
1
-2
2
-3
1
-11
-1
2
-1
8
I am a 1.
1
1
1
-2
2
-3
1
-11
-1
2
-1
8
I need to be 0.
I need to be 0.
-11 - 2(-2) 1 - 2(1) -3 - 2(1) 2 - 2(1)
1
1
1
-2
2
-3
1
-11
R2 2 R1
-1
2
-1
8
R3 R1
1
1
1
-2
-1 + 1 2+1 -1 + 1
8 + (-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?
1
1
1
-2
0
-5
-1
-7
0
3
0
6
1
1
1
-2
0
3
0
6
0
-5
-1
-7
To avoid fractions!
We will switch Row 2 and Row 3
I need to be 1
1
1
1
-2
0
3
0
6
0
-5
-1
-7
1
1
1
-2
0
1
0
2
0
-5
-1
-7
1 3
R2
I need to be 0.
I need to be 0.
1
1
1
-2
0
1
0
2
0
-5
-1
-7
1
0
1
-4
0
1
0
2
0
0
-1
3
R1 R2
R3 5 R2 0 + 5(0) -5 + 5(1) -1 + 5(0)
-7 + 5(2)
-2 - 2 1-0 1-1 1-0
I need to be 1
1
0
1
-4
0
1
0
2
0
0
-1
3
1
0
1
-4
0
1
0
2
0
0
1
-3
1 R3
I am a 0
I need to be 0.
1
0
1
-4
0
1
0
2
0
0
1
-3
1
0
0
-1
0
1
0
2
0
0
1
-3
R1 R3 -4 – (-3) 1-1 0-0 1-0
Reading the Solution 1
0
0
-1
x = -1
0
1
0
2
y=2
0
0
1
-3
z = -3
Writing the Solution x + y + z = -2 2x - 3y + z = -1 -11 -x + 2y - z = 8
1 3 6 2 1 5 4 8 X 9 4Y 3 4 5 7 9 1 1 2 3 2 X 0 3 Y 3 7 AX A X + BY = C DX + EY = F
AX A X + BY = C
DX + EY = F
AX A X = C - BY
DX = F - EY X = D-1(F – EY)
X = A-1(C – BY) X=X
A-1(C – BY) = D-1(F – EY) A-1C – – A A-1BY = D-1F – D-1EY D-1EY – – A A-1BY = D-1F – – A A-1C (D-1E – – A A-1B)Y = (D (D-1F – – A A-1C) Y = (D (D-1E – – A A-1B)-1(D-1F – – A A-1C)
AX A X + BY = C
DX + EY = F
BY = C - A AX X
EY = F - DX
Y = B-1(C – – A AX) X)
Y = E-1(F – DX)
Y = Y B-1(C – – A AX) X) = E-1(F – DX) B-1C – B-1 A AX X = E-1F – E-1DX E-1DX – B-1 A AX X = E-1F – B-1C (E-1D – B-1 A A)X )X = (E (E-1F – B-1C) X = (E (E-1D – B-1 A A))-1(E-1F – B-1C)
1 3 6 2 1 5 4 8 X 9 4Y 3 4 5 7 9 1 1 2 3 2 X 0 3 Y 3 7 7350 X = (E-1D – B-1 A)-1(E-1F – B-1C) X 9 73 Y = (D-1E – A-1B)-1(D-1F – A-1C)
77 219 Y 29 73
359 73
210 73
141 73
392 219
AX A X + BY = C
DX + EY = F
AX A X = C - BY X = A-1(C – BY)
D[ A-1(C – BY) ] + EY= F D A [ A-1C – – A A-1BY] + EY = F D A-1C – D A-1BY + EY = F EY – D A-1BY = F - D A-1C (E – D A-1B)Y = F - D A-1C Y = (E (E – D A-1B)-1(F - D A-1C)
AX A X + BY = C
DX + EY = F
BY = C - A AX X Y = B-1(C – – A AX) X)
DX + E [ B-1(C - AX ] = F DX + E [ B-1C - B-1 A AX X] =F DX + EB-1C - EB-1 A AX X =F DX - EB-1 A AX X = F - EB-1C (D - EB-1 A A)X )X = F - EB-1C X = (D (D - EB-1 A A))-1(F - EB-1C)
1 3 6 2 1 5 4 8 X 9 4Y 3 4 5 7 9 1 1 2 3 2 X 0 3 Y 3 7 7530 X 9 X = (D (D - EB-1 A A))-1(F - EB-1C) 73 Y = (E (E – D A-1B)-1(F - D A-1C)
27179 Y 2 9 73
359 73
2 1 0 73
141 73
3 9 2 219
ALJABAR MATRIKS AL (LANJUTAN) Ashar Saputra, PhD
Transpose Matriks Matriks •
Matriks Ortogonal •
Teori Dekomposisi Matriks Bila : [A] = sebuah matrix bujur sangkar =
…… ⋮ ⋮ ⋮ … ⋮ … 11
12
13
1
21
22
23
2
31
32
33
3
1
2
3
nxn
maka matrix tersebut dapat diekspresikan dalam bentuk: [A] = [L][U] dimana: 0 0 0 11
⋮ ⋮
21
[L] =
31
1
11
0
[L] =
0
0
⋮ ⋮
22 32
2
12 22
0
0
…… ⋮ … ⋮ … …… ⋮ … ⋮ 0
0
33
0
=
“lower triangle matrix”, matrix bujur sangkar yang semua elemen di atas/di kanan diagonal utama = 0
3
13
1
23
2
33
3
0
“upper triangle matrix”, matrix bujur sangkar yang = semua elemen di bawah/di kiri diagonal utama = 0
Teori Dekomposisi Matriks
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} dapat diperoleh tanpa INVERSE
{Y}
= [L]-1 . {B}
{X} = [U]-1 . ([L]-1 {B})
dapat diperoleh tanpa INVERSE
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
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 ..... ()
…… …… …… ⋮ ⋮ ⋮ … ⋮ ⋮ ⋮ ⋮ … ⋮ ⋮ ⋮ ⋮ … ⋮ 11
12
13
1
11
0
0
0
11
12
13
1
21
22
23
2
21
22
0
0
0
22
23
2
1
2
3
1
2
0
0
0
=
3
…… …… …… ⋮ ⋮ ⋮ … ⋮ ⋮ ⋮ ⋮ … ⋮ ⋮ ⋮ ⋮ … ⋮ ⋮ ⋮ −− ⋮ ⋮ − − − 11
12
13
1
11
0
0
0
11
12
13
1
21
22
23
2
21
22
0
0
0
22
23
2
1
2
3
1
2
0
0
0
=
3
sehingga:
2
11
=
11
21
=
11
.
1
=
11
.
11
=
21
21
=
1
1
=
11
21
11
1
11
22
=
21
.
21
+
22
.
22
21
=(
32
=
21
.
31
+
22
.
32
32
=
2
=
21
.
1
+
22
.
2
2
=
dan seterusnya:
3
=
3
21
32
21
21 . 31
22
31
2
21
.
1
.
2
22
.
1
33
32
2
1 2
)
Secara umum:
−− − − 1 =1
=
=
=0
1 =1
.
.
.
2
1 2
untuk i > j 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:
Bila:
= [L]
[M]
Maka: [A]
1
−
[A]
1
=
L L
=
L
=
L
T
1
T
1
1 T
. L
1
. L
1
1
= [M]T . M
L [L] 1 = [I] [L] [M] = [I]
dan
()
11
0
0
0
11
0
0
0
1 0 0
0
21
22
0
0
21
22
0
0
0 1 0
0
31
32
33
0
31
32
33
0
= 0 0 1
0
1
2
3
1
2
3
0 0 0
1
− ⋮ ⋮ − − − Sehingga: 11
.
21
.
31
.
11
=1
11
+
11
+
22
.
32
.
22
=0
21
+
33
.
31
11
=
21
=
=0
1
11
21
.
11
21
31
=
31
.
11 + 32
.
21
33
Secara umum:
=
1
1 =1
=
.
.
=0
untuk i > j 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:
11
[A] =
12
13
14
15
16
21
22
23
24
25
26
31
32
33
34
35
36
=
11
12
13
21
22
23
dimana:
11
=
11
;
12
=
21
21
=
31
;
22
=
12
13
14
22
23
24
32
33
34
....dst!
Aturan-aturan yang dipakai untuk mengoperasikan matrix partisi persis sama dengan mengoperasikan matrix biasa.
Contoh:
5 = 4 10 =
3 6 3
1 2 4
;
3 3
11
12
21
22 2 2
=
1 = 2 3
11
12
1
21
22
2
11
1
=
11
.
1
+
12
.
21
.
1
+
22
.
=
5 4
3 6
1 2
A12
B2 =
1 2
3
2
A21
B1 = 10
A22
B2 = 4
14 = 22
39 48
1 2
3
3
2
11 5 = 16 4
5 4
5 4 2
=
3 2
2
2
37 44
=
3 6
2 4
= 16
62
= 12
8
1
2 2 1
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
