514 Chapter 18Matrices and linear transformations
For order 2 the pair of inverse matrices is (ifad 1 − 1 bc 1 ≠ 10 )
(18.43)
EXAMPLE 18.15The inverse matrix of order 2
Show that if then
We have
and similarly forAA
− 1.
The denominator in the expression forA
− 1in (18.43) is the determinant of the
matrix A,
(18.44)
It follows that the inverse of the matrix Aexists only if the determinant of Ais not
zero. This is the general result. A square matrix with nonzero determinant is called
nonsingular. A matrix whose determinant is zero is called singular, and the inverse of
a singular matrix is not defined. The inverse of a nonsingular matrix can be obtained
by means of the following prescription.
1.Replace each element a
ijby its cofactor C
ijin the determinant of A(see
Section 17.2):
A=
aa a
aa a
aa a
nnn
nnn11
12121 2221
2
→
CC C
CC C
nn11
12121 222
CCC C
n
n1 nn
2detA==−
ab
cd
ad bc
AA
−=−
−
−
=−
−
−
11
2
42
31
12
34
1
2
20
022
10
01
=
A
−=−
−
−
11
2
42
31
A=
12
34
AA=
,=
−
−
−
−ab
cd
ad bc
db
ca
11