Next: Using Matrix Inverses Up: matrices3 Previous: Rules for Matrix Arithmetic

# Matrix Inverses

In an earlier handout, we hinted about the possibility of solving a matrix equation by multiplying both sides of the equation by an inverse to the matrix . Now we are going to define the inverse matrix and see how to compute it. First, we need to define another concept:

The identity matrix (sometimes denoted , if the dimension is not clear from context) is the matrix that has 1's down its main diagonal and 0's everyplace else:

The identity matrices are the 1's (the units'' in technical mathematical language) of matrix multiplication. That is, if is the identity matrix, and and are any matrices of the right dimensions so the following products are defined, then

Now an matrix has an inverse if there is an matrix such that

This means, in particular, that multiplying by is like dividing by :

Now that we have a definition of a matrix inverse, how do we find one? First, we need a small but important fact:

Fact: If and are matrices and , then also . (Remember that in general, , so the truth of this fact is not obvious.)

This fact means that if we can find an matrix with the property that , then we will know that also, so . And (guess what?) we know how to solve the matrix equation : Write down the augmented matrix , row-reduce it, then look at the equivalent matrix equation obtained from the new, row-reduced augmented matrix.

For example, let's try to find an inverse to the matrix

We try to solve the matrix equation by writing down the augmented matrix

and row-reducing it. (Some of these steps combine two operations in one. For example, the first step consists of first adding times row 1 to row 2 and then adding times row 1 to row 3. The second step is simply to multply row 2 by .)

This is the augmented matrix of the matrix equation

or

Therefore we have our solution:

This is one way things can work out when we try to find the inverse of a matrix A. Here's another: let us try to find an inverse to the matrix

We try to solve the matrix equation by writing down the augmented matrix

and row-reducing it:

This is the augmented matrix of the matrix equation

Now this equation has no solutions. How do we know this? The entries in the third row of the product

will be the products of the third row of the first factor with the columns of . Since the third row of the first factor consists entirely of zeroes, these products will all be zero as well, so whatever is, the third row of

will be ; it is impossible for it to be . Therefore, the matrix does not have an inverse.

These are the two possibilities. We can collect this information into a procedure:

To find the inverse of a square matrix , write down the matrix and then row-reduce it. Either it will row-reduce to a matrix of the form , in which case , or it will row-reduce to a matrix of the form where has a row consisting entirely of zeroes, in which case has no inverse.

Next: Using Matrix Inverses Up: matrices3 Previous: Rules for Matrix Arithmetic
Peter Kostelec
2000-05-08