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Properties of Matrix Operations E-Book   Chapter   Section Finding Inverse Matrices

Our main goal in this section is 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.

 

Definition 1  If A is a square matrix and we can find another matrix of the same size, say B, such that

                                                               

then we call A invertible and we say that B is an inverse of the matrix A.

 

If we can’t find such a matrix B we call A a singular 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.

 

Theorem 1 Suppose that A is invertible and that both B and C are inverses of A.  Then  and we will denote the inverse as .

 

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 .

Pf_Box

 

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.

 

Example 1  Given the matrix A verify that the indicated matrix is in fact the inverse.

                                    

Solution

To verify that we do in fact have the inverse we’ll need to check that

                                                            

This is easy enough to do and so we’ll leave it to you to verify the multiplication.

                                           

 

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.

 

Example 2  The matrix below does not have an inverse.

 

 

This is fairly simple to see.  If B has a matrix then it must be a  matrix.  So, let’s just take any old ,

                                                         

Now let’s think about the product BC.  We know that the 2nd row of BC can be found by looking at the following matrix multiplication,

                           

 

So, the second row of BC is , but if C is to be the inverse of B the product BC must be the identity matrix and this means that the second row must in fact be .

 

Now, C was a general  matrix and we’ve shown that the second row of BC is all zeroes and hence the product will never be the identity matrix and so B can’t have an inverse and so is a singular matrix.

 

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.

 

Definition 2  If A is a square matrix and  then,

                                                   

 

Example 3  Compute  for the matrix,

                                                              

Solution

From Example 1 we know that the inverse of A is,

                                                            

So, this is easy enough to compute.

                                    

 

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

 

Theorem 2 Suppose that A and B are invertible matrices of the same size. Then,

(a) AB is invertible and .

(b)  is invertible and .

(c) For   is invertible and .

(d) If c is any non-zero scalar then cA is invertible and  

(e)  is invertible and .

 

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.