In this section we are going to do a quick review of determinants and we’ll be concentrating almost exclusively on how to compute them.  For a more in depth look at determinants you should check out the second chapter which is devoted to determinants and their properties.  Also, we’ll acknowledge that the examples in this section are all examples that were worked in the second chapter.

 

We’ll start off with a quick “working” definition of a determinant.  See The Determinant Function from the second chapter for the exact definition of a determinant.  What we’re going to give here will be sufficient for what we’re going to be doing in this chapter.

 

So, given a square matrix, A, the determinant of A, denoted by ,  is a function that associated with A a number.  That’s it.  That’s what a determinant does.  It takes a matrix and associates a number with that matrix.  There is also some alternate notation that we should acknowledge because we’ll be using it quite a bit.  The alternate notation is, .

 

We now need to discuss how to compute determinants.  There are many ways of computing determinants, but most of the general methods can lead to some fairly long computations.  We will see one general method towards the end of this section, but there are some nice quick formulas that can help with some special cases so we’ll start with those.  We’ll be working mostly with matrices in this chapter that fit into these special cases.

 

We will start with the formulas for  and  matrices.

 

Definition 1  If  then the determinant of A is,

 

Definition 2  If  then the determinant of A is,

 

Okay, we said that these were “nice” and “quick” formulas and the formula for the  matrix is fairly nice and quick, but the formula for the  matrix is neither nice nor quick.  Luckily there are some nice little “tricks” that can help us to write down both formulas.

 

We’ll start with the following determinant of a  matrix and we’ll sketch in two diagonals as shown

Note that if you multiply along the green diagonal you will get the first product in formula for  matrices and if you multiply along the red diagonal you will get the second product in the formula.  Also, notice that the red diagonal, running from right to left, was the product that was subtracted off, while the green diagonal, running from left to right, gave the product that was added.

 

We can do something similar for  matrices, but there is a difference.  First, we need to tack a copy of the leftmost two columns onto the right side of the determinant.  We then have three diagonals that run from left to right (shown in green below) and three diagonals that run from right to left (shown in red below).

As will the  case, if we multiply along the green diagonals we get the products in the formula that are added in the formula and if we multiply long the red diagonals we get the products in the formula that are subtracted in the formula.

 

Here are a couple of quick examples.

 

Example 1  Compute the determinant of each of the following matrices. (a)    [Solution] (b)    [Solution] (c)    [Solution] Solution (a)   We don’t really need to sketch in the diagonals for  matrices.  The determinant is simply the product of the diagonal running left to right minus the product of the diagonal running from right to left.  So, here is the determinant for this matrix.  The only thing we need to worry about is paying attention to minus signs.  It is easy to make a mistake with minus signs in these computations if you aren’t paying attention.                                                  [Return to Problems]   (b)    Okay, with this one we’ll copy the two columns over and sketch in the diagonals to make sure we’ve got the idea of these down.   Now, just remember to add products along the left to right diagonals and subtract products along the right to left diagonals.                 [Return to Problems]   (c)   We’ll do this one with a little less detail.  We’ll copy the columns but not bother to actually sketch in the diagonals this time.                   [Return to Problems]

 

As we can see from this example the determinant for a matrix can be positive, negative or zero.  Likewise, as we will see towards the end of this review we are going to be especially interested in when the determinant of a matrix is zero.  Because of this we have the following definition.

 

Definition 3  Suppose A is a square matrix. (a) If  we call A a singular matrix. (b) If  we call A a non-singular matrix.

 

So, in Example 1 above, both A and B are non-singular while C is singular.

 

Before we proceed we should point out that while there are formulas for larger matrices (see the Determinant Function section for details on how to write them down) there are not any easy tricks with diagonals to write them down as we had for   and  matrices.

 

With the statement above made we should note that there is a simple formula for general matrices of certain kinds.  The following theorem gives this formula.

 

Theorem 1  Suppose that A is an  triangular matrix with diagonal entries , , … ,  the determinant of A is,

 

This theorem will be valid regardless of whether the triangular matrix is an upper triangular matrix or a lower triangular matrix.  Also, because a diagonal matrix can also be considered to be a triangular matrix Theorem 1 is also valid for diagonal matrices.

 

Here are a couple of quick examples of this.

 

Example 2  Compute the determinant of each of the following matrices.   Solution Here are these determinants.

 

There are several methods for finding determinants in general.  One of them is the Method of Cofactors.  What follows is a very brief overview of this method.  For a more detailed discussion of this method see the Method of Cofactors in the Determinants Chapter.

 

We’ll start with a couple of definitions first.

 

Definition 4  If A is a square matrix then the minor of , denoted by , is the determinant of the submatrix that results from removing the ith row and jth column of A.

 

Definition 5  If A is a square matrix then the cofactor of , denoted by , is the number .

 

Here is a quick example showing some minor and cofactor computations.

 

 

Example 3  For the following matrix compute the cofactors , , and .                                                        Solution In order to compute the cofactors we’ll first need the minor associated with each cofactor.  Remember that in order to compute the minor we will remove the ith row and jth column of A.   So, to compute  (which we’ll need for  ) we’ll need to compute the determinate of the matrix we get by removing the 1st row and 2nd column of A.  Here is that work.   We’ve marked out the row and column that we eliminated and we’ll leave it to you to verify the determinant computation.  Now we can get the cofactor.                                               Let’s now move onto the second cofactor.  Here is the work for the minor.   The cofactor in this case is,     Here is the work for the final cofactor.

 

Notice that the cofactor for a given entry is really just the minor for the same entry with a “+1” or a “-1” in front of it.  The following “table” shows whether or not there should be a “+1” or a “-1” in front of a minor for a given cofactor.

 

 

To use the table for the cofactor  we simply go to the i^th row and j^th column in the table above and if there is a “+” there we leave the minor alone and if there is a “-” there we will tack a “-1” onto the appropriate minor.  So, for  we go to the 3^rd row and 4^th column and see that we have a minus sign and so we know that .

 

Here is how we can use cofactors to compute the determinant of any matrix.

 

Theorem 2  If A is an  matrix. (a) Choose any row, say row i, then,                                                   (b) Choose any column, say column j, then,

 

Here is a quick example of how to use this theorem.

 

Example 4  For the following matrix compute the determinant using the given cofactor expansions.                                                            (a) Expand along the first row.   [Solution] (b) Expand along the third row.   [Solution] (c) Expand along the second column.   [Solution]   Solution First, notice that according to the theorem we should get the same result in all three parts.   (a) Expand along the first row.   Here is the cofactor expansion in terms of symbols for this part.                                                    Now, let’s plug in for all the quantities.  We will just plug in for the entries.  For the cofactors we’ll write down the minor and a “+1” or a “-1” depending on which sign each minor needs.  We’ll determine these signs by going to our “sign matrix” above starting at the first entry in the particular row/column we’re expanding along and then as we move along that row or column we’ll write down the appropriate sign.   Here is the work for this expansion.                           We’ll leave it to you to verify the  determinant computations. [Return to Problems]   (b) Expand along the third row.   We’ll do this one without all the explanations.                          So, the same answer as the first part which is good since that was supposed to happen.    Notice that the signs for the cofactors in this case were the same as the signs in the first case.  This is because the first and third row of our “sign matrix” are identical.  Also, notice that we didn’t really need to compute the third cofactor since the third entry was zero.  We did it here just to get one more example of a cofactor into the notes. [Return to Problems]   (c) Expand along the second column.   Let’s take a look at the final expansion.  In this one we’re going down a column and notice that from our “sign matrix” that this time we’ll be starting the cofactor signs off with a “-1” unlike the first two expansions.                          Again, the same as the first two as we expected. [Return to Problems]

 

As this example has show it doesn’t matter which row or column we expand along we will always get the same result.

 

In this example we performed a cofactor expansion on a  since we could easily check the results using the process we discussed above.  Let’s work one more example only this time we’ll find the determinant of a  matrix and so we’ll not have any choice but to use a cofactor expansion.

 

Example 5  Using a cofactor expansion compute the determinant of,   Solution Since the row or column to use for the cofactor expansion was not given in the problem statement we get to choose which one we want to use.  From the previous example we know that it won’t matter which row or column we choose..  However, having said that notice that if there is a zero entry we won’t need to compute the cofactor/minor for that entry since it will just multiply out to zero.   So, it looks like the second row would be a good choice for the expansion since it has two zeroes in it.  Here is the expansion for this row.  As with the previous expansions we’ll explicitly give the “+1” or “-1” for the cofactors and the minors as well so you can see where everything in the expansion is coming from.           We didn’t bother to write down the minors  and  because of the zero entry.  How we choose to compute the determinants for the first and last entry is up to us at this point.  We could use a cofactor expansion on each of them or we could use the technique we saw above.  Either way will get the same answer and we’ll leave it to you to verify these determinants.   The determinant for this matrix is,

 

We’ll close this review off with a significantly shortened version of Theorem 9 from Properties of Determinants section.  We won’t need most of the theorem, but there are two bits of it that we’ll need so here they are.  Also, there are two ways in which the theorem can be stated now that we’ve stripped out the other pieces and so we’ll give both ways of stating it here.

 

Theorem 3  If A is an  matrix then (a) The only solution to the system  is the trivial solution (i.e.  ) if and only if . (b) The system  will have a non-trivial solution (i.e.  ) if and only if .

 

Note that these two statements really are equivalent.  Also, recall that when we say “if and only if” in a theorem statement we mean that the statement works in both directions.  For example, let’s take a look at the second part of this theorem.  This statement says that if  has non-trivial solutions then we know that we’ll also have .  On the other hand, it also says that if  then we’ll also know that the system will have non-trivial solutions.

 

This theorem will be key to allowing us to work problems in the next section.

 

This is then the review of determinants.  Again, if you need a more detailed look at either determinants or their properties you should go back and take a look at the Determinant chapter.