Operations with Matrices Properties of Matrix Operations The Inverse of a Matrix

Mrs. Meena KumariDEPARTMENT OF ECONOMICSPGGCG-11 ,CHANDIGARH

MATRICES

nm

nmmnmmm

n

n

n

ij M

aaaa

aaaaaaaaaaaa

aA

][

321

3333231

2232221

1131211

Operations with Matrices

(i, j)-th entry:

ija

row: mcolumn: n

size: m×n

i-th row vector

iniii aaar 21 row matrix

j-th column vector

mj

j

j

j

c

cc

c2

1

Square matrix: m = n

Diagonal matrix:

),,,( 21 nddddiagA nn

n

M

d

dd

00

0000

2

1

Trace:

nnijaA ][ If

nnaaaATr 2211)(Then

Example:

654321

A

2

1

rr

,3211 r 6542 r

654321

A 321 ccc

,41

1

c ,

52

2

c

63

3c

Equal matrix:

nmijnmij bBaA ][ ,][ If

njmibaBA ijij 1 ,1 ifonly and if Then

Example: (Equal matrix)

dcba

BA 4321

BA If

4 ,3 ,2 ,1 Then dcba

Matrix addition:

nmijnmij bBaA ][ ,][ If

nmijijnmijnmij babaBA ][][][Then

Example : (Matrix addition)

3150

21103211

2131

1021

231

231

2233

11

000

Scalar multiplication:

scalar : ,][ If caA nmij

nmijcacA ][Then

Matrix subtraction:

BABA )1(

Ex 3: (Scalar multiplication and matrix subtraction)

212103421

A

231341002

B

Find (a) 3A, (b) –B, (c) 3A – B

Sol:(a)

212103421

33A

231323130333432313

636309

1263

(b)

231341002

1B

231341002

(c)

231341002

636309

12633 BA

4076410

1261

Matrix multiplication:

pnijnmij bBaA ][ ,][ If

pmijpnijnmij cbaAB ][][][Then

Size of ABwhere

njin

n

kjijikjikij babababac

1

2211

inijii

nnnjn

nj

nj

nnnn

inii

n

ccccbbb

bbbbbb

aaa

aaa

aaa

21

1

2221

1111

21

21

11211

BAAB

Example : Show that AB and BA are not equal for the matrices.

1231

A and

2012

B

Sol:

4452

2012

1231

AB

2470

1231

2012

BA

Note: Note: BAAB

Ex : (Find AB)

052431

A

1423

B

Sol:

)1)(0()2)(5()4)(0()3)(5()1)(2()2)(4()4)(2()3)(4()1)(3()2)(1()4)(3()3)(1(

AB

10156419

Three basic matrix operators: (1) matrix addition (2) scalar multiplication (3) matrix multiplication

Properties of Matrix Operations

Zero matrix:nm0

Identity matrix of order n:nI

(1) A+B = B + A

Properties of matrix addition and scalar multiplication:

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

(3) 1A = A

(4) c( A+B ) = cA + cB

calar cMA nm s: , If

Properties of zero matrices:

AA nm 0 (1)Then

nmA A 0)((2)

nmnm or A c cA 000)3(

Notes:

(1) 0m×n: the additive identity for the set of all m×n matrices(2) –A: the additive inverse of A

Transpose of a matrix:

nm

mnmm

n

n

M

aaa

aaaaaa

A

If

21

22221

11211

mn

mnnn

m

m

T M

aaa

aaaaaa

A

Then

21

22212

12111

Transpose of A matrix

(a)

82

A (b)

987654321

A (c)

114210

A

Sol: (a)

82

A 82 TA

(b)

987654321

A

963852741

TA

(c)(c)

114210

A

141

120TA

Properties of transposes:

AA TT )( )1(TTT BABA )( )2(

)()( )3( TT AccA

)( )4( TTT ABAB

Symmetric matrix:

A square matrix A is symmetric if A = AT

Skew-symmetric matrix:

A square matrix A is skew-symmetric if AT = –A

Example:

654321

Ifcb

aA is symmetric, find a, b, c?

Sol:

654321

cbaA

65342

1cba

AT

5 ,3 ,2 cba

TAA

Ex:

030210

Ifcb

aA is a skew-symmetric, find a, b, c?

Sol:

030210

cbaA

032

010

cba

AT

TAA 3 ,2 ,1 cba

The Inverse of a Matrix

The inverse of a matrix is unique.

(1) The inverse of A is denoted by

1A

IAAAA 11 )3(

AAA 111 )( and invertible is (2)

If A and B are invertible matrices of size n, then AB is invertible and

111)( ABAB