18.4 The inverse matrix 513
and the trace is
(18.40)
and this is identical to the sum in (18.39).
The transpose of a matrix product
The transpose of the product of two, or more, matrices is equal to the product of the
transpose matrices taken in reverse order,
(AB)
T1 = 1 B
TA
T(18.41)
EXAMPLE 18.14The transpose of a product
Let
Then
and
0 Exercise 46
18.4 The inverse matrix
If Aand Bare both squarematrices of order nthen Bis the inverse matrix of A(and
vice-versa) if
BA 1 = 1 AB 1 = 1 I
where Iis the unit matrix of order n. The inverse matrix of Ais denoted byA
− 1:
A
− 1A 1 = 1 AA
− 11 = 1 I (18.42)
BA
TT=
−
−−
21 0
32 1
42 2
10 0
21
01
32
=
=
43
97
22
21
()AB
TAB=
−
−
−
20 3
11 2
2341
1220
0120
=
4922
3721
AB=
−
−
,=
−
20 311 2
2341
1220
0120
tr trDBA==
==∑∑
kninki ikba
11