Understanding Engineering Mathematics

Before pressing on to the inverse matrix, let’s think about why we might need it and
what it might look like. Remember that we introduced the idea of a matrix so that we
could write a system of equations in the single form

Au=b

This is of little use unless we can push the new notation to help us to solve such a system.
Think of a simple equation

3 x= 6

Of course, the solution is x=2. But, spelling out the details, we first multiply by
1 / 3 = 3 −^1 ,

3 −^1 ( 3 x)=( 3 −^13 )x= 1 x

=x= 3 −^16 = 2

In order to repeat this for the matrix equation we needA−^1 , the equivalent of 3−^1 .This
motivates the following definition.
Theinverse,A−^1 , of a square matrixAis defined by

A−^1 A=I=AA−^1

whereIis the appropriate unit matrix. The inverse of a matrix exists only if the matrix is
non-singular(i.e. its determinant is non-zero). This follows because, since the determinant
of a product is the product of the determinants (390

➤

):

|A−^1 A|=|A−^1 ||A|=|I|= 1

and so we must have|A|=0. Also, from this equation follows:

|A−^1 |= 1 /|A|=|A|−^1

So
det(A−^1 )=[det(A)]−^1

In Problem 13.12 above we saw that

AAdjA=−I=|A|I

This is, in fact, a general result and leads to the following expression forA−^1 :

A−^1 =

AdjA

|A|

Note that in finding the inverse matrix it is wise to calculate|A|first, because if this is
zero, the inverse does not exist and it is a waste of time findingAdjA.