Our main goal in this section is to define inverse matrices and to take a look at some nice properties involving matrices.  We won’t actually be finding any inverse matrices in this section.  That is the topic of the next section. 

We’ll also take a quick look at elementary matrices which as we’ll see in the next section we can use to help us find inverse matrices.  Actually, that’s not totally true.  We’ll use them to help us devise a method for finding inverse matrices, but we won’t be explicitly using them to find the inverse.

So, let’s start off with the definition of the inverse matrix.

Note that we only talk about inverse matrices for square matrices.  Also note that if A is invertible it will on occasion be called non-singular.  We should also point out that we could also say that B is invertible and that A is the inverse of B.

Before proceeding we need to show that the inverse of a matrix is unique, that is for a given invertible matrix A there is exactly one inverse for the matrix.

Proof : Since B is an inverse of A we know that .  Now multiply both sides of this by C to get .  However, by the associative law of matrix multiplication we can also write  as .  Therefore, putting these two pieces together we see that  or .

So, the inverse for a matrix is unique.  To denote this fact we now will denote the inverse of the matrix A as  from this point on.

As the definition of an inverse matrix suggests, not every matrix will have an inverse.  Here is an example of a matrix without an inverse.

In the previous section we introduced the idea of matrix exponentiation.  However, we needed to restrict ourselves to positive exponents.  We can now take a look at negative exponents.

Next, let’s take a quick look at some nice facts about the inverse matrix.

Proof :

Note that in each case in order to prove that the given matrix is invertible all we need to do is show that the inverse is what we claim it to be.  Also, don’t get excited about showing that the inverse is what we claim it to be.  In these cases all we need to do is show that the product (both left and right product) of the given matrix and what we claim is the inverse is the identity matrix.  That’s it.

Also, do not get excited about the inverse notation.  For example, in the first one we state that .  Remember that the  is just the notation that we use to denote the inverse of AB.  This notation will not be used in the proof except in the final step to denote the inverse.

(a) Now, as suggested above showing this is not really all that difficult.  All we need to do is show that  and .  Here is that work.

So, we’ve shown both and so we now know that AB is in fact invertible (since we’ve found the inverse!) and that .

(b) Now, we know from the fact that A is invertible that

But this is telling us that if we multiply  by A on both sides then we’ll get the identity matrix.  But this is exactly what we need to show that  is invertible and that its inverse is A.

(c) The best way to prove this part is by a proof technique called induction.  However, there’s a chance that a good many of you don’t know that and that isn’t the point of this class.  Luckily, for this part anyway, we can at least outline another way to prove this.

To officially prove this part we’ll need to show that  .  We’ll show one of the inequalities and leave the other to you to verify since the work is pretty much identical.

Again, we’ll leave the second product to you to verify, but the work is identical.  After doing this product we can see that  is invertible and .

(d) To prove this part we’ll need to show that .  As with the last part we’ll do half the work and leave the other half to you to verify.

Upon doing the second product we can see that cA is invertible and .

(e) The part will require us to show that  and in keeping with tradition of the last couple parts we’ll do the first one and leave the second one to you to verify.

This one is a little tricky at first, but once you realize the correct formula to use it’s not too bad.  Let’s start with  and then remember that .  Using this fact (backwards) on  gives us,

Note that we used the fact that  here which we’ll leave to you to verify.

So, upon showing the second product we’ll have that  is invertible and .

Note that the first part of this theorem can be easily extended to more than two matrices as follows,

Now, in the previous section we saw that in general we don’t have a cancellation law or a zero factor property.  However, if we restrict ourselves just a little we can get variations of both of these.

Proof :

(a) Since we know that A is invertible we know that  exists so multiply on the left by  to get,

(b) Again we know that  exists so multiply on the left by  to get,

Note that this theorem only required that A be invertible, it is completely possible that the other matrices are singular. 

Note as well with the first one that we’ve got to remember that matrix multiplication is not commutative and so if we have  then there is no reason to think that  even if A is invertible.  Because we don’t know that  we’ve got to leave this as is.  Also when we multiply both sides of the equation by  we’ve got multiply each side on the left or each side on the right, which is again because we don’t have the commutative law with matrix multiplication.  So, if we tried the above proof on  we’d have,

In either case we don’t have .

Okay, it is now time to take a quick look at Elementary matrices.

Here are some examples of elementary matrices and the row operations that produced them.

Note that the fourth example above shows that any identity matrix is also an elementary matrix since we can think of arriving at that matrix by taking one times any row (not just the second as we used) of the identity matrix.

Here’s a really nice theorem about elementary matrices that we’ll be using extensively to develop a method for finding the inverse of a matrix.

Now, let’s go back to Example 4 for a second and notice that we can apply a second row operation to get the given elementary matrix back to the original identity matrix.

These kinds of operations are called inverse operations and each row operation will have an inverse operation associated with it.  The following table gives the inverse operation for each row operation.

Row operation	Inverse Operation
Multiply row i by	Multiply row i by
Interchange rows i and j	Interchange rows j and i
Add c times row i  to row j	Add  times row i  to row j

Now that we’ve got inverse operations we can give the following theorem.

Proof : This is actually a really simple proof.  Let’s start with .  We know from Theorem 4 that this is the same as if we’d applied the inverse operation to E, but we also know that inverse operations will take an elementary matrix back to the original identity matrix.  Therefore we have,

Likewise, if we look at  this will be the same as applying the original row operation to .  However, if you think about it this will only undo what the inverse operation did to the identity matrix and so we also have,

Therefore, we’ve proved that  and so E is invertible and .

Now, suppose that we’ve got two matrices of the same size A and B.  If we can reach B by applying a finite number of row operations to A then we call the two matrices row equivalent.  Note that this will also mean that we can reach A from B by applying the inverse operations in the reverse order.