The Inverse of a Matrix 7.4
Introduction
1
In number arithmetic every number a (6= 0) has a reciprocal b written as a−1 or such that
a
ba = ab = 1. Some, but not all, square matrices have inverses. If a square matrix A has an inverse,
A−1 , then
AA−1 = A−1 A = I.
We develop a rule for finding the inverse of a 2 × 2 matrix (where it exists) and we look at two
methods of finding the inverse of a 3 × 3 matrix (where it exists).
Non-square matrices do not possess inverses so this Section only refers to square matrices.
#
• be familiar with the algebra of matrices
Prerequisites • be able to calculate a determinant
Before starting this Section you should . . .
• know what a cofactor is
"
' !
$
• state the condition for the existence of an
inverse matrix
Learning Outcomes • use the formula for finding the inverse of
a 2 × 2 matrix
On completion you should be able to . . .
• find the inverse of a 3 × 3 matrix using row
operations and using the determinant method
& %
38 HELM (2008):
Workbook 7: Matrices
, ®
1. The inverse of a square matrix
We know that any non-zero number k has an inverse; for example 2 has an inverse 21 or 2−1 . The
inverse of the number k is usually written k1 or, more formally, by k −1 . This numerical inverse has
the property that
k × k −1 = k −1 × k = 1
We now show that an inverse of a matrix can, in certain circumstances, also be defined.
Given an n × n square matrix A, then an n × n square matrix B is said to be the inverse matrix
of A if
AB = BA = I
where I is, as usual, the identity matrix (or unit matrix) of the appropriate size.
Example 6
0 − 12
−1 1
Show that the inverse matrix of A = is B =
−2 0 1 − 12
Solution
All we need do is to check that AB = BA = I.
−1 1 1 0 −1 1 −1 1 0 −1 1 2 0 1 0
AB = ×2 =2 × =2 =
−2 0 2 −1 −2 0 2 −1 0 2 0 1
The reader should check that BA = I also.
We make three important remarks:
• Non-square matrices do not have inverses.
• The inverse of A is usually written A−1 .
• Not all square matrices have inverses.
Task
1 0 a b
Consider A = , and let B = be a possible inverse of A.
2 0 c d
(a) Find AB and BA:
Your solution
AB = BA =
HELM (2008): 39
Section 7.4: The Inverse of a Matrix
