linear algebra & matrices bachelor information technology dosen pengampu: asro nasiri drs, m.kom
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Linear Algebra & Matrices Bachelor Information Technology Dosen Pengampu: Asro Nasiri Drs, M.Kom. 2. Matrices & vector (1). STMIK AMIKOM YOGYAKARTA Jl. Ringroad Utara Condong Catur Yogyakarta. Telp. 0274 884201 Fax 0274-884208 Website: www.amikom.ac.id. matrices. Or. Row. Column. - PowerPoint PPT PresentationTRANSCRIPT
Linear Algebra & Matrices
Bachelor Information Technology
Dosen Pengampu:Asro Nasiri Drs, M.Kom.
STMIK AMIKOM YOGYAKARTAJl. Ringroad Utara Condong Catur Yogyakarta. Telp. 0274 884201 Fax 0274-884208Website: www.amikom.ac.id
2. Matrices & vector (1)
matrices
mnmm
n
n
aaa
aaa
aaa
A
21
22221
11211
mnmm
n
n
aaa
aaa
aaa
A
21
22221
11211
Or
mnmm
n
n
aaa
aaa
aaa
A
21
22221
11211
Row
Columnmatrices elementmatrices m x
nIf m = n is square matrices
VectorVector : a special matrices that only have
one row or one column. Row vector (one row) dan column vector
(one column)Contoh :
9
7
5
2
6
3
tor column vek
736
542 vektor row
dc
b
- a
• matrices A and B are equal if A and B have the same size and corresponding elements are equal.
• Vector A and B are equal if A and B have the same dimension and corresponding elements are equal.
C B C, A B, A
428
532
428
532
428
532
CBA
532
5
3
2
8
4
2
532
b
vua
a = b,
u ≠ v, a ≠ u ≠ v
and b ≠ u ≠ v
• matrices can also be referred as collection of vectors Amxn is matrices A that a collection of m row vector and n column vector.
428
4
5,
2
3,
8
2dan 53-2
: vectorfollowing theofconsist matrix a is 428
532
A
matrices and Vector Operation• Addition and substraction of matrices
two matrices can added and substrac if have same orde.A + B = C where cij = aij + bij
• Comutative law : A + B = B + A• Associative law : A + (B + C) = (A + B) + C =
A + B + C
Product of matrices with Scalar
• λA = B where bij = λaij
• example :
1815
126
6.35.3
4.32.33 So
3
65
42
BAA
A
matrices Product• matrices product of A x B is possible if
number of column of A equal the number of rows B.
• Amxn x Bnxp = Cmxp
5339
2317
8.47.36.45.3
8.27.16.25.1
86
75
43
21
Vector multiple of matrices
• Non vector matrices can be multiple with a column vector, if number of column is same with dimension of column vector. The result is a new column.
• Amxn x Bnx1 = Cmx1 n > 1
53
23
8.47.3
8.27.1
8
7
43
21
Special matrices
• Identity matrices : matrices square is if all element in main diagonal is 1, other diagonal are 0.
100
010
001
I 10
01I 32
matrices Diagonal
• matrices diagonal is square matrices that all element is zero except on main diagonal
10
01
400
030
003
50
03
matrices Identitas
matrices Null
• matrices null : matrices that all element are null 0
• Contoh :
000
0000
00
000 2x322x
Transpose matrices
• Row element transpose to column element vice versa
• Amxn=[aij] matrices transpose is A′nxm =[aji]
43
12'
41
32AA (A′) ′ = A
matrices Simetrik
• matrices simetrix if transpose same with its matrices.
• A = A′
73
31'
73
31AA
AA′ = AA = A2
skew symmetric
• A = -A atau A = -A′ ′
024
205
450
024
205
450
024
205
450
-A'A'A
inverse matrices
If a matrices when multiple of a square matrices resulting a identity matrices
A its inverse is A-1
AA-1 = IA-1 = adj.A |A|