Cambridge Additional Mathematics

326 Matrices (Chapter 12)

FURTHER MATRIX ALGEBRA

In this section we consider matrix algebra with inverse matrices.

Be careful that you use multiplication correctly. In particular,

remember that:

² We can only perform matrix multiplication if the orders of

the matrices allow it.

² If wepremultiply on one side then we mustpremultiply

on the other. This is important because, in general,

AB 6 =BA. The same applies if wepostmultiply.

Discovery 2 Properties of inverse matrices

#endboxedheading

In this Discovery, we consider some properties of invertible 2 £ 2 matrices.

What to do:

1 A matrixAisself-inversewhen A=A¡^1.

For example, if A=

μ

0 ¡ 1

¡ 10

¶

then A¡^1 =¡^11

μ

01

10

¶

=

μ

0 ¡ 1

¡ 10

¶

=A.

a Show that if A=A¡^1 , then A^2 =I.

b Show that there are exactly 4 self-inverse matrices of the form

μ

ab

ba

¶

.

2aGiven A=

μ

12

¡ 10

¶

, find A¡^1 and (A¡^1 )¡^1.

b IfAis any invertible matrix, simplify (A¡^1 )¡^1 (A¡^1 ) and (A¡^1 )(A¡^1 )¡^1 by replacing

A¡^1 by B.

c What can be deduced fromb?

3 Supposekis a non-zero number andAis an invertible matrix.

a Simplify (kA)(^1

k

A¡^1 ) and (^1

k

A¡^1 )(kA).

b What can you conclude from your results?

4aIf A=

μ

11

2 ¡ 1

¶

and B=

μ

01

2 ¡ 3

¶

, find in simplest form:

i A¡^1 ii B¡^1 iii (AB)¡^1

iv (BA)¡^1 v A¡^1 B¡^1 vi B¡^1 A¡^1

b Choose any two invertible matrices and repeata.

c What do the results ofaandbsuggest?

d Simplify (AB)(B¡^1 A¡^1 ) and (B¡^1 A¡^1 )(AB) given that A¡^1 and B¡^1 exist.

What can you conclude from your results?

Premultiply

Postmultiply

means multiply

on the left of each side.

means multiply

on the right of each side.

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Additional Mathematics
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