8.10 THE INVERSE OF A MATRIX
Find the inverse of the matrixA=
243
1 − 2 − 2
−33 2
.
We first determine|A|:
|A|=2[−2(2)−(−2)3] + 4[(−2)(−3)−(1)(2)] + 3[(1)(3)−(−2)(−3)]
=11. (8.58)This is non-zero and so an inverse matrix canbe constructed. To do this we need the
matrix of the cofactors,C, and henceCT. We find
C=
24 − 3
113 − 18
− 27 − 8
and CT=
21 − 2
4137
− 3 − 18 − 8
,
and hence
A−^1 =
CT
|A|
=
1
11
21 − 2
4137
− 3 − 18 − 8
. (8.59)
For a 2×2 matrix, the inverse has a particularly simple form. If the matrix isA=(
A 11 A 12
A 21 A 22)then its determinant|A|is given by|A|=A 11 A 22 −A 12 A 21 ,andthematrixof
cofactors is
C=(
A 22 −A 21
−A 12 A 11)
.Thus the inverse ofAis given by
A−^1 =CT
|A|=1
A 11 A 22 −A 12 A 21(
A 22 −A 12
−A 21 A 11). (8.60)
It can be seen that the transposed matrix of cofactors for a 2×2matrixisthe
same as the matrix formed by swapping the elements on the leading diagonal
(A 11 andA 22 ) and changing the signs of the other two elements (A 12 andA 21 ).
This is completely general for a 2×2 matrix and is easy to remember.
The following are some further useful properties related to the inverse matrix