Cambridge Additional Mathematics

314 Matrices (Chapter 12)

NEGATIVE MATRICES

ThenegativematrixA, denoted¡A, is actually¡ 1 A.

¡Ais obtained fromAby reversing the sign of each element ofA.

For example, if A=

μ

3 ¡ 1

24

¶

, then ¡A=

μ

¡ 1 £ 3 ¡ 1 £¡ 1

¡ 1 £ 2 ¡ 1 £ 4

¶

=

μ

¡ 31

¡ 2 ¡ 4

¶

The addition of a matrix and its negative always produces a zero matrix.

A+(¡A)=(¡A)+A=O

For example:

μ

3 ¡ 1

24

¶

+

μ

¡ 31

¡ 2 ¡ 4

¶

=

μ

00

00

¶

.

MATRIX ALGEBRA

We now compare our discoveries about matrices so far with ordinary algebra. We assume thatAandBare

matrices of the same order.

Ordinary algebra

² Ifaandbare real numbers then

a+b is also a real number.

² a+b=b+a

² (a+b)+c=a+(b+c)

² a+0=0+a=a

² a+(¡a)=(¡a)+a=0

² a half ofais

a

2

Matrix algebra

² IfAandBare matrices then

A+Bis a matrix of the

same order.

² A+B=B+A

² (A+B)+C=A+(B+C)

² A+O=O+A=A

² A+(¡A)=(¡A)+A=O

² a half ofAis^12 A

We always write

1

2 A and not

A

2

Example 5 Self Tutor

Show that:

a if X+A=B then X=B¡A b if 3 X=A then X=^13 A

a X+A=B

) X+A+(¡A)=B+(¡A)

) X+O=B¡A

) X=B¡A

b 3 X=A

)^13 (3X)=^13 A

) 1 X=^13 A

) X=^13 A

EXERCISE 12B.3

1 Simplify:

a A+2A b 3 B¡ 3 B c C¡ 2 C

d ¡B+B e 2(A+B) f ¡(A+B)

g ¡(2A¡C) h 3 A¡(B¡A) i A+2B¡(A¡B)

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_12\314CamAdd_12.cdr Tuesday, 7 January 2014 5:56:36 PM BRIAN