If A and B are square matrices such that
AB = I (the identity matrix),
then B is called the inverse of A and is denoted
by 
.
A square matrix A has either no inverse or a unique
inverse . In the first case, it is said to be
singular and in the second case nonsingular.
Interestingly, linear independence of vectors plays a
role here: a matrix is singular if its columns form a
set of linearly dependent vectors; and it is nonsingular
if its columns are linearly independent.
Another property is the following:
if B is the inverse of A, then A is the inverse of B.
An important property of nonsingular square matrices is the following. Consider the system of linear equations
simply written as Ax = b.
When A is a square nonsingular matrix,
this linear system has a unique solution, which can be obtained as follows.
Multiply the matrix equation Ax = b by on
the left:
This yields 
and so, the unique solution to the
system of linear equations is
Finding the Inverse of a Square Matrix
Given 
,
we must find 
such that AB = I (the identity matrix).
Therefore, the first column of B must satisfy 
(this vector is the 1st column of I). Similarly, for the other
columns of B. For example, the jth column of B satisfies
(the jth column of I). So
in order to
get the inverse of an 
matrix, we must solve n linear
systems. However, the same steps of the Gauss-Jordan elimination
procedure are
needed for all of these systems. So we solve them all at once,
using the matrix form.
Example: Find the inverse of 
.
We need to solve the following matrix equation
We divide the first row by 3 to introduce a 1 in the top left corner.
Then we add four times the first row to the second row to introduce a 0 in the first column.
Multiply the second row by 3.
Add 
the second row to the first. (All this is
classical Gauss-Jordan elimination.)
As IB = B, we get
It is important to note that, in addition to the two elementary row operations introduced earlier in the context of the Gauss-Jordan elimination procedure, a third elementary row operation may sometimes be needed here, namely permuting two rows.
Example: Find the inverse of 
.
Because the top left entry of A is 0, we need to permute rows 1 and 2 first.
Now we divide the first row by 2.
Next we add 
the second row to the first.
and we are done, since the matrix in front of B is the identity.
