TKE132102 Week 1 Aljabar-Matriks

Jurusan Teknik ElektroSemester Ganjil Tahun Ajaran 2015/2016

Pendidikan Untuk Indonesia Lebih Baik - Ari FadliPendidikan Untuk Indonesia Lebih Baik - Ari Fadli

Lecture #1Aljabar Matriks

Matematika TeknikTKE132102


Pendidikan Untuk Indonesia Lebih Baik - Ari Fadli

OutlineMatrix : Terminology, Types, Properties, Operation,

Determinan



Matrix - a rectangular array of

variables or constants in

horizontal rows and vertical

columns enclosed in brackets.

Element - each value in a matrix;

either a number or a constant.

Dimension - number of rows by

number of columns of a matrix.

Matrix Terminology



Column Matrix - a matrix with only one column.

Row Matrix - a matrix with only one row.

Square Matrix - a matrix that has the same number of rows and columns.

Equal Matrices - two matrices that have the same dimensions

and each element of one matrix is equal to the corresponding

element of the other matrix

Zero matrices - Every element of a matrix is zero, it is called a zero

matrix, i.e.,

Types of Matrices



upper triangular matrix lower triangular matrix

4000

6300

5220

1341

4125

0323

0022

0001

identity matrix

1000

0100

0010

0001

Types of Matrices

diagonal matrix

4000

0300

0020

0001

Symmetric AT = A

653

542

321

orthogonal AAT = ATA



A + B = B + A

A + (B +C) = (A + B) +C

l(A + B) = lA + lB, where l is a scalar

AB BA in general

(commutative law)

(associative law)

(distributive law)

Properties of matrices



Matrices Operations

Skalar dan matriks



Determinant - Every square matrix has associated with it a scalarcalled its determinant (det(A) or |A|)

**To find a determinant you must have

a SQUARE MATRIX!!**

Determinant Matrix Terminology



Determinants from Matrices and Transpose have the same value

If one row/cols of a matrix consists entirely of zeros, det = 0

2624

37

A 26

23

47

A

0

567

000

312

A 0

087

056

022

A

Properties

If two rows of a matrix are interchanged, the det changes sign.

3132

57

A 31

57

32

A



If two cols of a matrix are identical, the determinant is zero.

027

27

A 0

303

232

111

A

Properties

If the matrix B is obtained from the matrix A by multiplying every element in

one row of A by the scalar , then |B|= |A|.

543

12

A 357

2821

12

AB

For an n n matrix A and any scalar , det(A)= ndet(A).

543

12

A 20

86

24

A 22 X 5 = 20



If a matrix B is obtained from a matrix A by adding to one row of A, a scalar

times another row of A, then |A|=|B|.

The determinant of a triangular matrix, either upper or lower, is the product of

the elements on the main diagonal.

If A and B are of the same order, then det(AB)=det(A) det(B).

543

12

A 5

21

12

B

15

500

310

273

A

Properties

If det(A) = 0. Thus, A is not invertible

6 42

1 0 1

3 2 1

A



Sarrus

Minor and Coffactor

Elementary Row Operations (OBE)

Properties



then,A

dc

babcadAdet

h

e

b

g

d

a

ihg

fed

cba

A

bdi- afh -ceg- cdh bfg aeiAdet then ,

Properties



Given a matrix A=[aij] , the cofactor of the element aij is a scalar obtained

by multiplying together the term (-1)i+j and the minor obtained from A by

removing the ith row and the jth column. In other words, the cofactor Cij is

given by Cij = (1)i+jMij.

333231

232221

131211

aaa

aaa

aaa

A3332

1312

21aa

aaM

3331

1311

22aa

aaM

2121

12

21 )1( MMC

2222

22

22 )1( MMC

Expansi Cofactor



To find the determinant of a matrix A of arbitrary order,

Pick any one row or any one column of the matrix; For each element in the row or column chosen, find its cofactor; Multiply each element in the row or column chosen by its cofactor and

sum the results. This sum is the determinant of the matrix.

ininiiii

n

jijij CaCaCaCaAA

22111

)det(

njnjjjjj

n

iijij CaCaCaCaAA

22111

)det(

ith row expansion

jth column expansion

Expansi Cofactor



333231

232221

131211

aaa

aaa

aaa

A

333231

232221

131211

MMM

MMM

MMM

MINOR

333231

232221

131211

CCC

CCC

CCC

COFACTOR

332313

322212

312111

CCC

CCC

CCC

cofactorADJ T

Expansi Cofactor



3620

4211

4521

6242

A

Rows Reduction

3620

4211

4521

3121

A

3620

1310

1600

3121

A

3620

1600

1310

3121

A

5000

1600

1310

3121

A2(-1)(1)(6)(5)



cols Reduction

3620

4211

4521

6242

A

3000

4103/51

4133/21

61002

A

3000

4103/50

4133/25/3

61002

A

3000

41000

4132/35/3

6103/52

A

3000

41000

4132/30

61033/4

A

3000

4103/50

4133/25/3

61002

A

TKE132102 Week 1 Aljabar-Matriks

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