Modern Control Engineering

Note that the element of the jth row and ith column of the product A(adjA)is

Hence,A(adjA)is a diagonal matrix with diagonal elements equal to , or

A(adjA)= I

Similarly, the element in the jth row and ith column of the product (adjA)Ais

Hence, we have the relationship

A(adjA)=(adjA)A= I (C–1)

Thus

whereAijis the cofactor of aijof the matrix A. Thus, the terms in the ith column of A^1

are l/ times the cofactors of the ith row of the original matrix A. For example, if

then the adjoint of Aand the determinant @A@are respectively found to be

A= C

1 2 0

3 - 1 - 2

1 0- 3

S

@A@

G

A 11

@A@

A 21

@A@

p

An 1

@A@

A 12

@A@

A 22

@A@

p

An 2

@A@

oo o

A 1 n

@A@

A 2 n

@A@

p

Ann

@A@

A W

• (^1) =adjA

@A@

=

@A@

a

n

k= 1

bjkaki= a

n

k= 1

Akjaki=dij@A@

@A@

@A@

a

n

k= 1

ajkbki= a

n

k= 1

ajkAik=dji@A@

Appendix C / Vector-Matrix Algebra 877

adjA=G W

= C

3 6- 4

7 - 3 2

1 2- 7

S

`

3 - 1

1 0

-

12

10

` `

1 2

3 - 1

`

• `

3 - 2

1 - 3

` `

1 0

1 - 3

-

1 0

3 - 2

`

`

- 1 - 2

0 - 3

-

2 0

0 - 3

` `

2 0

- 1 - 2

`