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)
cyan magenta yellow black
(^05255075950525507595)
100 100
(^05255075950525507595)
100 100 4037 Cambridge
Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_12\314CamAdd_12.cdr Tuesday, 7 January 2014 5:56:36 PM BRIAN