Cambridge Additional Mathematics

Matrices (Chapter 12) 323

The real numbers 5 and^15 are calledmultiplicative inversesbecause when they are multiplied together, the

result is the multiplicative identity 1 : 5 £^15 =^15 £5=1

For the matrices

μ

25

13

¶

and

μ

3 ¡ 5

¡ 12

¶

, we notice that

μ

25

13

¶μ

3 ¡ 5

¡ 12

¶

=

μ

10

01

¶

=I

and

μ

3 ¡ 5

¡ 12

¶μ

25

13

¶

=

μ

10

01

¶

=I.

We say that

μ

25

13

¶

and

μ

3 ¡ 5

¡ 12

¶

aremultiplicative inversesof each other.

Themultiplicative inverseofA, denotedA¡^1 , satisfies AA¡^1 =A¡^1 A=I.

To find the multiplicative inverse of a matrixA, we need a matrix which, when multiplied byA, gives the

identity matrixI.

We will now determine how to find the inverse of a matrixA.

Suppose A=

μ

ab

cd

¶

and A¡^1 =

μ

wx

yz

¶

) AA¡^1 =

μ

ab

cd

¶μ

wx

yz

¶

=I

)

μ

aw+by ax+bz

cw+dy cx+dz

¶

=

μ

10

01

¶

)

½

aw+by=1 .... (1)

cw+dy=0 .... (2)

and

½

ax+bz=0 .... (3)

cx+dz=1 .... (4)

Solving (1) and (2) simultaneously forwandygives: w=

d

ad¡bc

and y=

¡c

ad¡bc

.

Solving (3) and (4) simultaneously forxandzgives: x=

¡b

ad¡bc

and z=

a

ad¡bc

.

So, if A=

μ

ab

cd

¶

where ad¡bc 6 =0, then A¡^1 =

1

ad¡bc

μ

d ¡b

¡ca

¶

.

In this case A¡^1 A=

1

ad¡bc

μ

d ¡b

¡ca

¶μ

ab

cd

¶

=

1

ad¡bc

μ

ad¡bc bd¡bd

ac¡ac ¡bc+ad

¶

=

μ

10

01

¶

=I also,

so A¡^1 A=AA¡^1 =I

D The inverse of a × matrix

4037 Cambridge

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