Cambridge Additional Mathematics

Premultiplymeans

multiply on the left

of each side.

Matrices (Chapter 12) 327

From theDiscoveryyou should have found that ifAandBare invertible, then:

² (A¡^1 )¡^1 =A ² (kA)¡^1 =

1

k

A¡^1 ² (AB)¡^1 =B¡^1 A¡^1

Example 12 Self Tutor

If A^2 =2A+3I, find A¡^1 in the linear form rA+sI whererandsare scalars.

A^2 =2A+3I

) A¡^1 A^2 =A¡^1 (2A+3I) fpremultiplying both sides byA¡^1 g

) A¡^1 AA=2A¡^1 A+3A¡^1 I

) IA=2I+3A¡^1

) A¡ 2 I=3A¡^1

) A¡^1 =^13 (A¡ 2 I)

) A¡^1 =^13 A¡^23 I

EXERCISE 12D.2

1 Suppose A=

μ

21

01

¶

, B=

μ

12

¡ 10

¶

, C=

μ

03

12

¶

, and AXB=C. FindX.

2 Suppose X,Y, andZ are 2 £ 1 matrices, andAandBare invertible 2 £ 2 matrices.

If X=AY and Y=BZ, write:

a Xin terms ofZ b Zin terms ofX.

3 If A=

μ

32

¡ 2 ¡ 1

¶

, write A^2 in the linear form pA+qI wherepandqare scalars.

Hence write A¡^1 in the form rA+sI whererandsare scalars.

4 Write A¡^1 in linear form given that:

a A^2 =4A¡I b 5 A=I¡A^2 c 2 I=3A^2 ¡ 4 A

5 It is known that AB=A and BA=B where the matricesAandBare not necessarily invertible.

Prove that A^2 =A.

Hint: From AB=A, you cannot deduce that B=I.

6 Under what condition is it true that “if AB=AC then B=C”?

7 If X=P¡^1 AP and A^3 =I, prove that X^3 =I.

8 If aA^2 +bA+cI=O and X=P¡^1 AP, prove that aX^2 +bX+cI=O.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_12\327CamAdd_12.cdr Tuesday, 7 January 2014 6:09:32 PM BRIAN