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|