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# Inverse of a Square Matrix

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.

Next: Determinants Up: Basic Linear Algebra Previous: Linear Combinations

Michael A. Trick
Mon Aug 24 16:30:59 EDT 1998