5.3. Determinants via Row Reduction#
In this section we will first consider the effect of row operations on the value of a determinant.
This leads the way to a more efficient way to compute
It also leads the way to two very important properties of determinants, namely
-
The product rule:
. -
The matrix
is invertible if and only if .
5.3.1. How Row Operations affect a Determinant#
We have seen in Section 5.2 that the cofactor expansion of an
Proposition 5.3.1 (How row operations affect a determinant)
For the determinant of an
-
If a row of
is scaled with a factor , the determinant is scaled with a factor . -
If a multiple of one row of
is added to another row, the determinant does not change. -
When two rows of
are swapped, the determinant changes sign.
We postpone the proof until the end of this section and first look at examples and a few consequences.
Example 5.3.1
The following identities show what happens with a
-
the second row is scaled with a factor
: -
the first row is added
times to the third row: -
the first and the fourth row are swapped:
Note that these properties can be expressed using elementary matrices (cf. Section 3.2) .
Example 5.3.2
Let
Then we have
Since
we see that in all three cases we have that
Since every row operation can be performed using the product with an elementary matrix,
a consequence of Proposition 5.3.1 is that Equation (5.3.1) holds for any product of an elementary
These are the basics for the general product rule we will see later, which states that
for arbitrary
The next two examples illustrate the practical use of the three rules involving row operations.
Example 5.3.3
The steps involved are:
- (1) take out a factor
from the first row, - (2) subtract the first row
times from the second row and times from the third row (or: add it times and times respectively) - (3) expand along the first column.
Evaluating the
Can you describe the row operations and cofactor expansions in the following computation?
Remark 5.3.1
Because of Proposition 5.2.3 that states
every rule involving row operations may be transformed into a rule about column operations. It is here that computing a determinant differs strikingly from the reduction of a (for instance augmented) matrix to an echelon matrix. Another, more subtle difference is that
a row operation applied to a matrix leads to an equivalent matrix, which we denote by the symbol
Note that in Rule i. of Proposition 5.3.1 the factor
In the next example column operations are used.
Example 5.3.4
And do you see what is happening here?
An interesting consequence of rule (3) of Proposition 5.3.1 is the following.
Corollary 5.3.1
If a matrix
Proof of Corollary 5.3.1
Suppose the
On the one hand,
on the other hand, because of Proposition 5.3.1, Rule 2, we have
We may conclude
5.3.2. Determinants versus Invertibility#
With the knowledge built so far we can show the important property that was already hinted at in Section 5.2.
Theorem 5.3.1
For any square matrix
The proof is – we think – quite instructive. (However, feel free to skip it.)
Proof of Theorem 5.3.1
In the previous section we have already seen that the statement is true for triangular matrices.
Now suppose
From Proposition 5.2.2 we know that for a triangular matrix
‘
The row operations transforming
We have seen (Equation (5.3.1) in Example 5.3.2) that
Furthermore, the determinant of an elementary matrix is nonzero. Namely, for a row scaling it is equal to
with
So if
We conclude that
Theorem 5.3.2
For two
The idea of the proof is to break it down to products of the form
Proof of Theorem 5.3.2
We already know that the identity holds if
Hence suppose that the matrix
So then, step by step we find that
and also
Corollary 5.3.2
If the matrix
Proof of Corollary 5.3.2
We can combine the three properties
i.
as follows:
so indeed
Exercise 5.3.1
For each of the following statements decide whether they are true or false. In case true, give an argument, in case false, give a counterexample.
-
For each
matrix it holds that
-
For each two
matrices and it holds that
-
For each
matrix it holds that
-
For each
matrix and each real number it holds that
Solution to Exercise 5.3.1
We treat the statements one by
-
Is it true that for each
matrix it holds that ?This is true, and follows from repeatedly using the property
. Namely,
-
Is it true that for each two
matrices and it holds that
This statement is false. A trivial counterexample is given by
, for . Namely, for these matrices we see that
-
Is it true that for each
matrix and each real number it holds that
This is true. One way to prove it is to write
, where
So we find
-
Is it true that
for each matrix ?This is not true in general. Taking
in the previous statement we see that
A specific example: for
it holds that
We will conclude this section, for the interested reader, with a proof of the properties of Proposition 5.3.1. In fact we will prove the column version, and we add one related rule that will be of use both immediately in the proof and also later on.
Proposition 5.3.2
Suppose
Then
Click on the symbol to the right below for the proof of Proposition 5.3.1 and Proposition 5.3.2.
Proof of Proposition 5.3.1 and Proposition 5.3.2
For typographical reasons we will prove the three rules stated as column operations.
For an
the rules can then be formulated as
- (1)
; - (2)
; - (3)
.
So, let us consider them one by one.
- (1) If a column is scaled with a factor
, then the determinant is also scaled with a factor .
Suppose
Then expanding det
If we take out the constant factor
Proposition 5.3.2 is proved in much the same way as rule (1) by expansion along the
- (2) The proof of the swapping rule is more involved.
One way to set about is
-
first settle the rule when the first two columns are interchanged;
-
next consider the swapping of two arbitrary consecutive columns;
-
finally note that any column swap is the result of an odd number of swaps of consecutive columns.
We will consider the swapping of the first two columns in complete detail. Let
To make this explicit for a
Expanding det
Noting that
The same argument works for the interchanging of two arbitrary consecutive columns.
And the argument can even be generalized for two columns that are not necessarily neighbours. The notation with many indices becomes hard to read though. As stated the swapping of two arbitrary columns can be accomplished via an odd number of ‘consecutive swaps’, so then the determinant changes sign an odd number of times.
And for an odd number
In fact, to swap columns
Lastly we have to prove
- (3) If the multiple of one column of
is added to another column, the determinant does not change.
First Rule (2) implies, as in Corollary 5.3.1, that a determinant with two equal columns has the value 0.
We then proceed as follows for Rule (3):
This settles all matters.
5.3.3. Grasple Exercises#
Grasple Exercise 5.3.1
Effects of row operations on a 3x3 determinant
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Grasple Exercise 5.3.2
Effects of row operations on a 3x3 determinant
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Grasple Exercise 5.3.3
Effects of row and column operations on a 3x3 determinant
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Grasple Exercise 5.3.4
Effects of several operations on a 4x4 determinant
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Grasple Exercise 5.3.5
Effects of a column operation on a 4x4 determinant
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Grasple Exercise 5.3.6
Effects of a column operation on a 4x4 determinant
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Grasple Exercise 5.3.7
To compute a 3x3 determinant using row reduction
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Grasple Exercise 5.3.8
To compute a 4x4 determinant with quite a few zeros
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Grasple Exercise 5.3.9
To compute a 4x4 determinant via reduction and expansion
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Grasple Exercise 5.3.10
To compute a ‘random’ 5x5 determinant with entries in {-2,-1,0,1,2}
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Grasple Exercise 5.3.11
Computing a structured 5x5 determinant in a ‘smart’ way
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Grasple Exercise 5.3.12
Finding a parameter
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Grasple Exercise 5.3.13
Checking linear (in)dependence of
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Grasple Exercise 5.3.14
Checking linear (in)dependence of
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Grasple Exercise 5.3.15
Checking invertibility of a matrix
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Grasple Exercise 5.3.16
Find
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Grasple Exercise 5.3.17
To find det
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Grasple Exercise 5.3.18
To combine several rules of determinants for a product involving three matrices
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Grasple Exercise 5.3.19
To find det
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Grasple Exercise 5.3.20
To find det
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Grasple Exercise 5.3.21
What can det(
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Grasple Exercise 5.3.22
What about det(
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Grasple Exercise 5.3.23
(True/False) det
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Grasple Exercise 5.3.24
What happens to det(A) if the last column of
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Grasple Exercise 5.3.25
What happens to det(
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At the end a non-Grasple exercise.
Exercise 5.3.2
Give an alternative proof of Corollary 5.3.1 using Rule i. and Rule ii. of Proposition 5.3.1.
Solution to Exercise 5.3.2
Suppose
If we subtract the
For instance, with a
Expansion of the last determinant across its fourth row yields the value