You can navigate through this E-Book using the menu to the left. For E-Books that have a Chapter/Section organization each option in the menu to the left indicates a chapter and will open a menu showing the sections in that chapter. Alternatively, you can navigate to the next/previous section or chapter by clicking the links in the boxes at the very top and bottom of the material.
Also, depending upon the E-Book, it will be possible to download the complete E-Book, the chapter containing the current section and/or the current section. You can do this be clicking on the E-Book, Chapter, and/or the Section link provided below.
For those pages with mathematics on them you can, in most cases, enlarge the mathematics portion by clicking on the equation. Click the enlarged version to hide it.
We’ll start off the chapter by defining the determinant
function. This is not such an easy thing
however as it involves some ideas and notation that you probably haven’t run
across to this point. So, before we
actually define the determinant function we need to get some preliminaries out
of the way.
First, a permutation
of the set of integers 
is an arrangement of all the integers in the
list without omission or repetitions. A
permutation of will typically be denoted by where is the first number in the permutation, is the second number in the permutation, etc.
|
Example 1 List
all permutations of .
Solution
This one isn’t too bad because there are only two integers
in the list. We need to come up with
all the possible ways to arrange these two numbers. Here they are.
|
|
Example 2 List
all the permutations of
Solution
This one is a little harder to do, but still isn’t too
bad. We need all the arrangements of
these three numbers in which no number is repeated or omitted. Here they are.
|
From this point on it can be somewhat difficult to find
permutations for lists of numbers with more than 3 numbers in it. One way to make sure that you get all of them
is to write down a permutation tree. Here is the permutation tree for .
At the top we list all the numbers in the list and from this
top number we’ll branch out with each of the remaining numbers in the
list. At the second level we’ll again
branch out with each of the numbers from the list not yet written down along
that branch. Then each branch will
represent a permutation of the given list of numbers
As you can see the number of permutations for a list will
quick grow as we add numbers to the list.
In fact it can be shown that there are n! permutations of the list ,
or any list containing n distinct
numbers, but we’re going to be working with so that’s the one we’ll reference. So, the list will have permutations, the list will have permutations, etc.
Next we need to discuss inversions in a permutation. An inversion
will occur in the permutation whenever a larger number precedes a smaller
number. Note as well we don’t mean that
the smaller number is immediately to the right of the larger number, but
anywhere to the right of the larger number.
|
Example 3 Determine
the number of inversions in each of the following permutations.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
Solution
(a)
Okay, to count the number of inversions we will start at
the left most number and count the number of numbers to the right that are
smaller. We then move to the second
number and do the same thing. We
continue in this fashion until we get to the end. The total number of inversions are then the
sum of all these.
We’ll do this first one in detail and then do the
remaining ones much quicker. We’ll
mark the number we’re looking at in red and to the side give the number of
inversions for that particular number.
In the first case there are two numbers to the right of 3
that are smaller than 3 so there are two inversions there. In the second case we’re looking at the
smallest number in the list and so there won’t be any inversions there. Then with 4 there is one number to the
right that is smaller than 4 and so we pick up another inversion. There is no reason to look at the last
number in the permutation since there are no numbers to the right of it and
so won’t introduce any inversions.
The permutation has a total of 3 inversions.
[Return to Problems]
(b)
We’ll do this one much quicker. There are inversions in . Note that each number in the sum above
represents the number of inversion for the number in that position in the
permutation.
[Return to Problems]
(c)
There are inversions in .
[Return to Problems]
(d)
There are no inversions in .
[Return to Problems]
(e)
There are in .
[Return to Problems]
|
Next, a permutation is called even if the number of inversions is even and odd if the number of inversions is odd.
|
Example 4 Classify
as even or odd all the permutations of the following lists.
(a)
(b)
Solution
(a) Here’s a
table giving all the permutations, the number of inversions in each and the
classification.
|
Permutation
|
# Inversions
|
Classification
|
|
|
0
|
even
|
|
|
1
|
odd
|
(b) We’ll do the
same thing here
|
Permutation
|
# Inversions
|
Classification
|
|
|
0
|
even
|
|
|
1
|
odd
|
|
|
1
|
odd
|
|
|
2
|
even
|
|
|
2
|
even
|
|
|
3
|
odd
|
We’ll need these results later in the section.
|
Alright, let’s move back into matrices. We still have some definitions to get out of
the way before we define the determinant function, but at least we’re back
dealing with matrices.
Suppose that we have an matrix, A,
then an elementary product from this
matrix will be a product of n entries
from A and none of the entries in the
product can be from the same row or column.
|
Example 5 Find
all the elementary products for,
(a) a
matrix
[Solution]
(b) a
matrix.
[Solution]
Solution
(a) a matrix.
Okay let’s first write down the general matrix.
Each elementary product will contain two terms and since
each term must come from different rows we know that each elementary product
must have the form,
All we need to do is fill in the column subscripts and
remember in doing so that they must come from different columns. There are really only two possible ways to
fill in the blanks in the product above.
The two ways of filling in the blanks are and and yes we did mean to use the permutation
notation there since that is exactly what we need. We will fill in the blanks with all the
possible permutations of the list of column numbers, in this case.
So, the elementary products for a matrix are
[Return to Problems]
(b) a matrix.
Again, let’s start off with a general matrix for reference purposes.
Each of the elementary products in this case will involve
three terms and again since the must all come from different rows we can
again write down the form they must take.
Again, each of the column subscripts will need to come
from different columns and like the case we can get all the possible choices for
these by filling in the blanks will all the possible permutations of .
So, the elementary products of the are,
[Return to Problems]
|
A general matrix A,
will have n! elementary products of
the form
where ranges over all the permutations of .
We can now take care of the final preliminary definition
that we need for the determinant function.
A signed elementary product
from A will be the elementary product
that is multiplied by “+1” if is an even permutation or multiplied by “-1”
if is an odd permutation.
|
Example 6 Find
all the signed elementary products for,
(a) a
matrix
[Solution]
(b) a
matrix.
[Solution]
Solution
We listed out all the elementary products in Example 5 and
we classified all the permutations used in them as even or odd in Example
4. So, all we need to do is put all
this information together for each matrix.
(a) a matrix.
Here are the signed elementary products for the matrix.
|
Elementary
Product
|
Permutation
|
Signed Elementary
Product
|
|
|
- even
|
|
|
|
- odd
|
|
[Return to Problems]
(b) a matrix.
Here are the signed elementary products for the matrix.
[Return to Problems]
|
Okay, we can now give the definition of the determinant
function.
|
Definition 1 If
A is square matrix then the determinant function is denoted by det and det(A) is defined to be the
sum of all the signed elementary products of A.
|
Note that often we will call the number det(A) the determinant of A. Also, there is some
alternate notation that is sometimes used for determinants. We will sometimes denote determinants as and this is most often done with the actual
matrix instead of the letter representing the matrix. For instance for a matrix A
we will use any of the following to denote the determinant,
So, now that we have the definition of the determinant
function in hand we can actually start writing down some formulas. We’ll give the formula for and matrices only because for any matrix larger
than that the formula becomes very long and messy and at those sizes there are
alternate methods for computing determinants that will be easier.
So, with that said, we’ve got all the signed elementary
products for and matrices listed in Example 6 so let’s write
down the determinant function for these matrices.
First the determinant function for a matrix.
Now the determinant
function for a matrix.
Okay, the formula for a matrix isn’t too bad, but the formula for a is messy and would not be fun to
memorize. Fortunately, there is an easy
way to quickly “derive” both of these formulas.
Before we give this quick trick to “derive” the formulas we
should point out that what we’re going to do ONLY works for and matrices.
There is no corresponding trick for larger matrices!
Okay, let’s start with a matrix.
Let’s examine the determinant below.
Notice the two diagonals that we’ve put on this
determinant. The diagonal that runs from
left to right also covers the positive elementary product in the formula. Likewise, the diagonal that runs from right
to left covers the negative elementary product.
So, for a matrix all we need to do is write down the
determinant, sketch in the diagonals multiply along the diagonals then add the
product if the diagonal runs from left to right and subtract the product if the
diagonal runs from right to left.
Now let’s take a look at a matrix.
There is a similar trick that will work here, but in order to get it to
work we’ll first need to tack copies the first 2 columns onto the right side of
the determinant as shown below.
With the addition of the two extra columns we can see that
we’ve got three diagonals running in each direction and that each will cover
one of the elementary products for this matrix.
Also, the diagonals that run from left to right cover the positive
elementary products and those that run from right to left cover the negative
elementary product. So, as with the matrix, we can quickly write down the
determinant function formula here by simply multiplying along each diagonal and
then adding it if the diagonal runs left to right or subtracting it if the
diagonal runs right to left.
Let’s take a quick look at a couple of examples with numbers
just to make sure we can do these.
|
Example 7 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 this example has shown determinants of matrices can be
positive, negative or zero.
It is again worth noting that there are no such tricks for
computing determinants for matrices larger that
In the remainder of this chapter we’ll take a look at some
properties of determinants, two alternate methods for computing them that are
not restricted by the size of the matrix as the two quick tricks we saw in this
section were and an application of determinants.