Cambridge Additional Mathematics

322 Matrices (Chapter 12)

4aIf A=

à 1

2

1

2

1

2

1

2

!

, determine A^2.

b Comment on the following argument for a 2 £ 2 matrixAsuch that A^2 =A:

A^2 =A

) A^2 ¡A=O

) A(A¡I)=O

) A=O or A¡I=O

) A=O or I

c Findall 2 £ 2 matricesAfor which A^2 =A. Hint: Let A=

μ

ab

cd

¶

.

5 Giveoneexample which shows that “if A^2 =O then A=O”isafalsestatement.

Example 9 Self Tutor

For A=

μ

12

34

¶

, find constantsaandbsuch that A^2 =aA+bI.

Since A^2 =aA+bI,

μ

12

34

¶μ

12

34

¶

=a

μ

12

34

¶

+b

μ

10

01

¶

)

μ

1+6 2+8

3 + 12 6 + 16

¶

=

μ

a 2 a

3 a 4 a

¶

+

μ

b 0

0 b

¶

)

μ

710

15 22

¶

=

μ

a+b 2 a

3 a 4 a+b

¶

Thus a+b=7 and 2 a=10

) a=5 and b=2

Checking for consistency:

3 a= 3(5) = 15 X 4 a+b= 4(5) + (2) = 22 X

6 Find constantsaandbsuch that A^2 =aA+bI, given:

a A=

μ

12

¡ 12

¶

b A=

μ

31

2 ¡ 2

¶

7aFor A=

μ

12

¡ 1 ¡ 3

¶

, find constantspandqsuch that A^2 =pA+qI.

b Hence, writeA^3 in the linear form rA+sI whererandsare scalars.

c WriteA^4 in linear form.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_12\322CamAdd_12.cdr Tuesday, 7 January 2014 5:57:36 PM BRIAN