8.10 THE INVERSE OF A MATRIX
Evaluate the determinant|A|=
∣
∣
∣∣
∣
∣∣
1023
01 − 21
3 − 34 − 2
− 21 − 2 − 1
∣
∣
∣∣
∣
∣∣
.
Taking a factor 2 out of the third column and then adding the second column to the third
gives
|A|=2
∣
∣∣
∣
∣
∣∣
1013
01 − 11
3 − 32 − 2
− 21 − 1 − 1
∣
∣∣
∣
∣
∣∣
=2
∣
∣∣
∣
∣
∣∣
1013
0101
3 − 3 − 1 − 2
−21 0− 1
∣
∣∣
∣
∣
∣∣
.
Subtracting the second column from the fourth gives
|A|=2
∣
∣∣
∣
∣∣
∣
1013
0100
3 − 3 − 11
−21 0− 2
∣
∣∣
∣
∣∣
∣
.
We now note that the second row has only one non-zero element and so the determinant
may conveniently be writtenas a Laplace expansion, i.e.
|A|=2× 1 ×(−1)2+2
∣
∣
∣∣
∣
∣
113
3 − 11
− 20 − 2
∣
∣
∣∣
∣
∣
=2
∣
∣
∣∣
∣
∣
404
3 − 11
− 20 − 2
∣
∣
∣∣
∣
∣
,
where the last equality follows by adding the second row to the first. It can now be seen
that the first row is minus twice the third, and so the value of the determinant is zero, by
property (v) above.
8.10 The inverse of a matrixOur first use of determinants will be in defining theinverseof a matrix. If we
were dealing with ordinary numbers we would consider the relationP=ABas
equivalent toB=P/A, provided thatA= 0. However, ifA,BandPare matrices
then this notation does not have an obvious meaning. What we really want to
know is whether an explicit formula forBcan be obtained in terms ofAand
P. It will be shown that this is possible for those cases in which|A|=0.A
square matrix whose determinant is zero is called asingularmatrix; otherwise it
isnon-singular. We will show that ifAis non-singular we can define a matrix,
denoted byA−^1 and called theinverseofA, which has the property that ifAB=P
thenB=A−^1 P.Inwords,Bcan be obtained by multiplyingPfrom the left by
A−^1. Analogously, ifBis non-singular then, by multiplication from the right,
A=PB−^1.
It is clear thatAI=A ⇒ I=A−^1 A, (8.53)whereIis the unit matrix, and soA−^1 A=I=AA−^1. These statements are