Cambridge Additional Mathematics

Matrices (Chapter 12) 321

Example 7 Self Tutor

Expand and simplify where possible:

a (A+2I)^2 b (A¡B)^2

a (A+2I)^2

=(A+2I)(A+2I) fX^2 =XX by definitiong

=(A+2I)A+(A+2I)2I fB(C+D)=BC+BDg

=A^2 +2IA+2AI+4I^2 f(C+D)B=CB+DBg

=A^2 +2A+2A+4I fAI=IA=A and I^2 =Ig

=A^2 +4A+4I

b (A¡B)^2

=(A¡B)(A¡B) fX^2 =XX by definitiong

=(A¡B)A¡(A¡B)B fC(D¡E)=CD¡CEg

=A^2 ¡BA¡AB+B^2 f(D¡E)C=DC¡ECg

Example 8 Self Tutor

If A^2 =2A+3I, findA^3 andA^4 in the linear form kA+lI wherekandlare scalars.

A^3 =A£A^2

=A(2A+3I)

=2A^2 +3AI

= 2(2A+3I)+3AI

=7A+6I

A^4 =A£A^3

=A(7A+6I)

=7A^2 +6AI

= 7(2A+3I)+6A

=20A+21I

EXERCISE 12C.3

1 Given that all matrices are 2 £ 2 andIis the identity matrix, expand and simplify:

a A(A+I) b (B+2I)B c A(A^2 ¡ 2 A+I)

d A(A^2 +A¡ 2 I) e (A+B)(C+D) f (A+B)^2

g (A+B)(A¡B) h (A+I)^2 i (3I¡B)^2

2aIf A^2 =2A¡I, findA^3 andA^4 in the linear form kA+lI wherekandlare scalars.

b If B^2 =2I¡B, findB^3 ,B^4 , andB^5 in linear form.

c If C^2 =4C¡ 3 I, findC^3 andC^5 in linear form.

3aIf A^2 =I, simplify:

i A(A+2I) ii (A¡I)^2 iii A(A+3I)^2

b If A^3 =I, simplify A^2 (A+I)^2.

c If A^2 =O, simplify:

i A(2A¡ 3 I) ii A(A+2I)(A¡I) iii A(A+I)^3

bcannot be simplified further

since, in general,AB BA 6 =.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_12\321CamAdd_12.cdr Wednesday, 8 January 2014 11:29:05 AM BRIAN