506 Chapter 18Matrices and linear transformations
EXAMPLES 18.6Multiplication by a scalar
If
then
0 Exercise 19
It follows from the rules of addition and multiplication by a scalar that a
linear combination of m 1 × 1 nmatrices is an m 1 × 1 nmatrix whose elements are the
linear combinations of corresponding elements. If A 1 = 1 (a
ij),B 1 = 1 (b
ij), and C 1 = 1 (c
ij)
then
αA 1 + 1 βB 1 + 1 γC 1 = 1 (αa
ij1 + 1 βb
ij1 + 1 γc
ij) (18.23)
where α, β, and γare scalars.
EXAMPLES 18.7Linear combinations of matrices
0 Exercise 20
Them 1 × 1 nmatrix whose elements are all zero is called them 1 × 1 nnull matrix(or zero
matrix) 0. It follows from the above rules that
(18.24)
if αβ thenβ
α
AB+= A=− B
0
2
11
20
23
22
40
46
−
−
−
−
−
==
00
00
00
2
123
456
3
120
301
110 6
110
−
−
−
=
−
− 115
−=
−−
−
,=−
−
AA
12
30
51
3
36
90
15 3
,=
0
00
00
00
A
A=−
−
12
30
51