512 Chapter 18Matrices and linear transformations
EXAMPLE 18.12Multiplication by a unit matrix
0 Exercise 43
The product is a zero matrix
IfAB 1 = 10 then it does notnecessarily follow thatA 1 = 10 orB 1 = 10 or thatBA 1 = 10 (even
if it exists).
EXAMPLE 18.13The product is a zero matrix
0 Exercises 44, 45
The determinant and trace of a matrix product
If Aand Bare square matrices of the same order then the determinant of the product
AB, and of BA, is equal to the product of the determinants of Aand B,
det 1 AB 1 = 1 det 1 A 1 × 1 det 1 B 1 = 1 det 1 BA (18.38)
IfC 1 = 1 ABthen, by the prescription (18.26), a diagonal element of the product is
The trace of the product matrix is therefore
(18.39)
This is equal to the trace of the product in reverse order. Thus if D 1 = 1 BAthen a
diagonal element of Dis
dba
kkinki ik=
=∑
1tr trCAB==
==∑∑
inknik kiab
11cab
iiknik ki=
=∑
110
10
00
11
00
00
00
1
=
but
11
10
10
00
20
=
100
010
001
23
12
01
23
−
= 112
01
23
12
01
10
01
−
=
−