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 *j*th column of *B* satisfies
(the *j*th 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.

Mon Aug 24 13:24:14 EDT 1998