Cambridge Additional Mathematics

320 Matrices (Chapter 12)

b Now let A=

μ

ab

cd

¶

, B=

μ

pq

rs

¶

, and C=

μ

wx

yz

¶

.

Prove that in general, A(B+C)=AB+AC.

c Use the matrices you ‘made up’ inato verify that (AB)C=A(BC).

d Prove that (AB)C=A(BC).

5aIf

μ

ab

cd

¶μ

wx

yz

¶

=

μ

ab

cd

¶

, show that w=z=1 and x=y=0 is a

solution for any values ofa,b,c, andd.

b For any real numbera, we know that a£1=1£a=a.

Is there a matrixIsuch that AI=IA=A for all 2 £ 2 matricesA?

6 Suppose A^2 =AA=A£A and that A^3 =AAA.

a FindA^2 if A=

μ

21

3 ¡ 2

¶

. b FindA^3 if A=

μ

5 ¡ 1

24

¶

.

c If A=

0

@

12

34

56

1

A try to findA^2.

d Under what conditions can we square a matrix?

7 Show that if I=

μ

10

01

¶

then I^2 =I and I^3 =I.

In theDiscoveryyou should have found that:

Note that in general, A(kB)=k(AB) 6 =kBA. We can change the order in which we multiply by a scalar,

but we cannot reverse the order in which we multiply matrices.

I=

μ

10

01

¶

is called the

identity matrix.

Ordinary algebra

² Ifaandbare real numbers then

so isab. fclosureg

² ab=ba for alla,b fcommutativeg

² a0=0a=0 for alla

² ab=0, a=0or b=0

² a(b+c)=ab+ac

fdistributive lawg

² a£1=1£a=a fidentity lawg

² an exists for all a> 0 and n 2 R.

Matrix algebra

² IfAandBare matrices that can be multiplied

thenABis also a matrix. fclosureg

² In general AB 6 =BA. fnon-commutativeg

² IfOis a zero matrix then

AO=OA=O for allA.

² ABmay beOwithout requiring

A=O or B=O.

² A(B+C)=AB+AC fdistributive lawg

² IfIis theidentity matrix

μ

10

01

¶

then

AI=IA=A for all 2 £ 2 matricesA.

fidentity lawg

² An exists providedAis square and n 2 Z+.

fNull Factor lawg

cyan magenta yellow black

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_12\320CamAdd_12.cdr Wednesday, 8 January 2014 11:28:59 AM BRIAN