Section 2.6 – The Inverse of a Square Matrix
1
Section 2.6  The Inverse of a Square Matrix
In this section we discuss a procedure for finding the inverse of a matrix and show how
the inverse can be used to help us solve a system of linear equations.
Let A be a square matrix of size n.
A square matrix
1

A
of size n such that
n
I
A
A
AA
=
=


1
1
is called the
inverse of A
.
Note:
Not every square matrix has an inverse.
A matrix with no inverse is called
singular
.
Finding the Inverse of a Matrix
Given the nxn matrix
A
:
1.
Adjoin the nxn identity matrix
I
to obtain the augmented matrix
(
I
A

2.
Use a sequence of row operations to reduce
(
I
A

to the form
(
B
I

, if possible.
The matrix
B
is the inverse of
A
.
Matrices That Have No Inverses
If there is a row to the left of the vertical line in the augmented matrix containing all
zeros, then the matrix does not have an inverse.
Example 1:
Find the inverse of
a.
=
4
3
2
1
A
.
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Section 2.6 – The Inverse of a Square Matrix
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b.
B =
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