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Online Notes / Linear Algebra / Determinants / Properties of Determinants
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The Determinant Function E-Book   Chapter   Section The Method of Cofactors

In this section we’ll be taking a look at some of the basic properties of determinants and towards the end of this section we’ll have a nice test for the invertibility of a matrix.  In this section we’ll give a fair number of theorems (and prove a few of them) as well as examples illustrating the theorems.  Any proofs that are omitted are generally more involved than we want to get into in this class.

 

Most of the theorems in this section will not help us to actually compute determinants in general.  Most of these theorems are really more about how the determinants of different matrices will relate to each other.  We will take a look at a couple of theorems that will help show us how to find determinants for some special kinds of matrices, but we’ll have to wait until the next two sections to start looking at how to compute determinants in general.

 

All of the determinants that we’ll be computing in the examples in this section will be of a  or a  matrix.  If you need a refresher on how to compute determinants of these kinds of matrices check out this example in the previous section.  We won’t actually be showing any of that work here in this section.

 

Let’s start with the following theorem.

 

Theorem 1  Let A be an  matrix and c be a scalar then,

 

 

Proof : This is a really simply proof.  From the definition of the determinant function in the previous section we know that the determinant is the sum of all the signed elementary products for the matrix.  So, for cA we will sum signed elementary products that are of the form,

 

 

 

Recall that for scalar multiplication we multiply all the entries by c and so we’ll have a c on each entry as shown above.  Also, as shown, we can factor all n of the c’s out and we’ll get what we’ve shown above.   Note that  is the signed elementary product for A. 

 

Now, if we add all the signed elementary products for cA  we can factor the  that is on each term out of the sum and what we’re left with is the sum of all the signed elementary products of A, or in other words, we’re left with det(A).  So, we’re done.

Pf_Box

 

Here’s a quick example to verify the results of this theorem.

 

Example 1  For the given matrix below compute both det(A) and det(2A).

                                                          

Solution

We’ll leave it to you to verify all the details of this problem.  First the scalar multiple.

                                                      

The determinants.

                          

 

Now, let’s investigate the relationship between det(A), det(B) and det(A+B).  We’ll start with the following example.

 

Example 2  Compute det(A), det(B) and det(A+B) for the following matrices.

                                      

Solution

Here all the determinants.

                       

 

Notice here that for this example we have .  In fact this will generally be the case. 

 

There is a very special case where we will get equality for the sum of determinants, but it doesn’t happen all that often.  Here is the theorem detailing this special case.

 

Theorem 2  Suppose that A, B, and C are all  matrices and that they differ by only a row, say the kth row.  Let’s further suppose that the kth row of C can be found by adding the corresponding entries from the kth rows of A and B.  Then in this case we will have that

                                                    

 

The same result will hold if we replace the word row with column above.

 

Here is an example of this theorem.

 

Example 3  Consider the following three matrices.

                  

 

First, notice that we can write C as,

                               

 

All three matrices differ only in the second row and the second row of C can be found by adding the corresponding entries from the second row of A and B.

 

The determinants of these matrices are,

              

 

Next let’s look at the relationship between the determinants of matrices and their products.

 

Theorem 3  If A and B are matrices of the same size then

                                                     

 

This theorem can be extended out to as many matrices as we want.  For instance,

 

 

 

Let’s check out an example of this.

 

Example 4  For the given matrices compute det(A), det(B), and det(AB).

                            

Solution

Here’s the product of the two matrices.

                                                         

Here are the determinants.

                                   

 

Here is a theorem relating determinants of matrices and their inverse (provided the matrix is invertible of course…).

 

Theorem 4  Suppose that A is an invertible matrix then,

                                                          

 

Proof : The proof of this theorem is a direct result of the previous theorem.  Since A is invertible we know that .  So take the determinant of both sides and then use the previous theorem on the left side.

 

 

Now, all that we need is to know that  which you can prove using Theorem 8 below.

 

 

Pf_Box

 

Here’s a quick example illustrating this.

 

Example 5  For the given matrix compute det(A) and .

                                                              

Solution

We’ll leave it to you to verify that A is invertible and that its inverse is,

                                                           

Here are the determinants for both of these matrices.

                                    

 

The next theorem that we want to take a look at is a nice test for the invertibility of matrices.

 

Theorem 5  A square matrix A is invertible if and only if .  A matrix that is invertible is often called non-singular and a matrix that is not invertible is often called singular.

 

Before doing an example of this let’s talk a little bit about the phrase “if and only if” that appears in this theorem.  That phrase means that this is kind of like a two way street.  This theorem, because of the “if and only if” phrase, says that if we know that A is invertible then we will have .  If, on the other hand, we know that  then we will also know that A is invertible.

 

Most theorems in this presented in these notes are not “two way streets” so to speak.  They only work one way, if however, we do have a theorem that does work both ways you will always be able to identify it by the phrase “if and only if”.

 

Now let’s work an example to verify this theorem.

 

Example 6  Compute the determinants of the following two matrices.

                            

Solution

We determined the invertibility of both of these matrices in the section on Finding Inverses so we already know what the answers should be (at some level) for the determinants.  In that section we determined that C was invertible and so by Theorem 5 we know that the det(C) should be non-zero.  We also determined that B was singular (i.e. not invertible) and so we know by Theorem 5 that det(B) should be zero.