Modern Control Engineering

and

=17

Hence, the inverse of Ais

In what follows, we give formulas for finding inverse matrices for the 2*2matrix

and the 33matrix. For the 22matrix

the inverse matrix is given by

For the 3*3matrix

the inverse matrix is given by

A= C

abc

def

gh i

S where@A@ Z 0

A-^1 =

1

ad-bc

c

d -b

- ca

d

A= c

ab

cd

d wheread-bcZ 0

A-^1 =

adjA

@A@

= C

3

17

6

17 -

4

17

7

17 -

3

17

2

17

1

17

2

17 -

7

17

S

@A@

878 Appendix C / Vector-Matrix Algebra

G W

`

de

gh

-

ab

gh

` `

ab

de

`

• `

df

gi

` `

ac

gi

-

ac

df

A-^1 = `

1

@A@

`

ef

hi

-

bc

hi

` `

bc

ef

`

Note that

There are several more useful formulas available. Assume that A=n*nmatrix,

B=nmmatrix,C=mnmatrix, and D=m*mmatrix. Then

[A+BC]-^1 =A-^1 - A-^1 B[Im+CA-^1 B]-^1 CA-^1

(A-^1 )=(A)-^1

(A-^1 )¿=(A¿)-^1

(A-^1 )-^1 =A

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