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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.
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.
Here’s a quick example to verify the results of this
theorem.
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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.
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Now, let’s investigate the relationship between det(A), det(B) and det(A+B).
We’ll start with the following example.
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Example 2 Compute
det(A), det(B) and det(A+B) for the following matrices.
Solution
Here all the determinants.
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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.
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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.
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Here is an example of this theorem.
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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,
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Next let’s look at the relationship between the determinants
of matrices and their products.
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Theorem 3 If
A and B are matrices of the same size then
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This theorem can be extended out to as many matrices as we
want. For instance,
Let’s check out an example of this.
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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.
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Here is a theorem relating determinants of matrices and
their inverse (provided the matrix is invertible of course…).
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Theorem 4 Suppose
that A is an invertible matrix
then,
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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.
Here’s a quick example illustrating this.
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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.
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The next theorem that we want to take a look at is a nice
test for the invertibility of matrices.
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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.
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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 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.
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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.
Here are those determinants of these two matrices.
Sure enough we got zero where we should have and didn’t
get zero where we should have.
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Here is a theorem relating the determinants of a matrix and
its transpose.
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Theorem 6 If
A is a square matrix then,
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Here is an example that verifies the results of this
theorem.
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Example 7 Compute
det(A) and for the following matrix.
Solution
We’ll leave it to you to verify that
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There are a couple special cases of matrices that we can
quickly find the determinant for so let’s take care of those at this point.
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Theorem 7 If
A is a square matrix with a row or
column of all zeroes then
and so A will be
singular.
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Proof : The proof
here is fairly straight forward. The
determinant is the sum of all the signed elementary products and each of these
will have a factor from each row and a factor from each column. So, in particular it will have a factor from
the row or column of all zeroes and hence will have a factor of zero making the
whole product zero.
All of the products are zero and upon summing them up we
will also get zero for the determinant.
Note that in the following example we don’t need to worry
about the size of the matrix now since this theorem gives us a value for the
determinant. You might want to check the
and to verify that the determinants are in fact
zero. You also might want to come back
and verify the other after the next section where we’ll learn methods for
computing determinants in general.
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Example 8 Each
of the following matrices are singular.
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It is actually very easy to compute the determinant of any
triangular (and hence any diagonal) matrix.
Here is the theorem that tells us how to do that.
So, what this theorem tells us is that the determinant of
any triangular matrix (upper or lower) or any diagonal matrix is simply the
product of the entries from the matrices main diagonal.
We won’t do a formal proof here. We’ll just give a quick outline.
Proof Outline : Since
we know that the determinant is the sum of the signed elementary products and
each elementary products has a factor from each row and a factor from each
column because of the triangular nature of the matrix, the only elementary
product that won’t have at least one zero is . All the others will have at least one zero in
them. Hence the determinant of the
matrix must be
Let’s take the determinant of a couple of triangular
matrices. You should verify the and matrices and after the next section come back
and verify the other.
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Example 9 Compute
the determinant of each of the following matrices.
Solution
Here are these determinants.
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We have one final theorem to give in this section. In the Finding Inverse section we gave a theorem that listed
several equivalent statements. Because
of Theorem 5 above we can add a statement to that theorem so let’s do that.
Here is the improved theorem.