Linear Algebra 10

66 CHAPTER 2. DETERMINANTS
2.2 Properties of Determinants
In this section, we will study properties determinants have and we will see how
these properties can help in computing the determinant of a matrix. We will
also see how these properties can give us information about matrices.
2.2.1 Determinants and Elementary Row Operations
We study how performing an elementary row operation on a matrix a¤ects its
determinant. This, in turn, will give us a powerful tool to compute determinants.
We give the main result as a theorem. Its proof will be given at the end of the
section.
Theorem 151 Let A and B be an n  n matrix.
1. If B is obtained by replacing one row of A by itself plus a multiple of
another row, then jBj = jAj.
2. If B is obtained by multiplying a row of A by a nonzero constant k, then
jBj = k jAj.
3. If B is obtained by interchanging two rows of A, then jBj = ??jAj.
We illustrate the theorem for 3  3 matrices. Assuming that the original
matrix we have is

a11 a12 a13
a21 a22 a23
a31 a32 a33

, we see what happens to its determinant
as we perform one of the elementary row operations.
Relationship Operation 
ka11 ka12 ka13
a21 a22 a23
a31 a32 a33

= k

a11 a12 a13
a21 a22 a23
a31 a32 a33

(kR1) ! (R1)

a21 a22 a23
a11 a12 a13
a31 a32 a33

= ??

a11 a12 a13
a21 a22 a23
a31 a32 a33

(R1) ! (R2)

a11 + ka21 a12 + ka22 a13 + ka23
a11 a12 a13
a31 a32 a33

=

a11 a12 a13
a21 a22 a23
a31 a32 a33

(R1 + kR2) ! (R1)
In particular, looking at the ?rst row of this table, we see that we can "factor"
a constant from any row.
This theorem is very important for computing determinants. recall from
the previous section that the determinant of a triangular matrix is the product
of the entries on its diagonal. A matrix in row-echelon form is a triangular
matrix. So, a strategy to compute the determinant of a matrix is to transform
the matrix into a row-echelon matrix using elementary row transformations,
recording how these elementary row transformations a¤ect the determinant of
2.2. PROPERTIES OF DETERMINANTS 67
the matrix. More speci?cally, if A is a matrix and U a row-echelon form of A
then
jAj = (??1)r  jUj (2.2)
where r is the number of times we performed a row interchange and  is the
product of all the constants k which appear in row operations of the form
(kRi) ! (Ri).
We illustrate this with a few examples.
Example 152 Compute jAj for A =
2
4
1 ??4 2
??2 8 ??9
??1 7 0
3
5.
The strategy is to reduce A into row-echelon form and use the fact that the
determinant of a triangular matrix is the product of the diagonal entries.

1 ??4 2
??2 8 ??9
??1 7 0

=

1 ??4 2
0 0 ??5
0 3 2

= ??

1 ??4 2
0 3 2
0 0 ??5

= ??(1) (3) (??5)
= 15
On the ?rst line, we performed (R2 + 2R1) ! (R2) and (R3 + R1) ! (R3).
These two transformations do not change the determinant. On the second line,
we switched rows 2 and 3, this introduces the minus sign we see. On the third
line, we simply used the fact that the determinant of a triangular matrix is the
product of the diagonal entries.
Example 153 Find jAj for A =
2
664
2 ??8 6 8
3 ??9 5 10
??3 0 1 ??2
1 ??4 0 6
3
775
.
We proceed as above.

2 ??8 6 8
3 ??9 5 10
??3 0 1 ??2
1 ??4 0 6

= 2

1 ??4 3 4
3 ??9 5 10
??3 0 1 ??2
1 ??4 0 6

= 2

1 ??4 3 4
0 3 ??4 ??2
0 ??12 10 10
0 0 ??3 2

= 2

1 ??4 3 4
0 3 ??4 ??2
0 0 ??6 2
0 0 ??3 2

68 CHAPTER 2. DETERMINANTS
= 2

1 ??4 3 4
0 3 ??4 ??2
0 0 ??6 2
0 0 0 1

= 2 (1) (3) (??6) (1)
= ??36
On the ?rst line, we factored out 2 from the ?rst row. On line 2, we performed
(R2 ?? 3R1) ! (R2), (R3 + 3R1) ! (R3) and (R4 ?? R1) ! (R4). These trans-
formations do not change the determinant. On line 3, we performed (R3 + 4R2) ! (R3), again this  leaves the determinant unchanged. On line 4, we performed
R4 ??
1
2
R3

! (R4) which, again, ;eaves the determinant unchanged. Once
we have a triangular matrix, we compute its determinant by multiplying the
diagonal entries.
Remark 154 In the above examples, we actually did not obtain a row-echelon
matrix. According to our de?nition, the ?rst nonzero entry of each row also
called a pivot element, should have been a 1. Doing this simply requires a trans-
formation of the form (kRi) ! (Ri). But as we can see, it is not necessary. In
fact, even what we did on the ?rst line of the above example, factoring the 2,
was not necessary. It simply made our computations easier. For the purpose of
computing the determinant of a matrix A, we only need to transform it into a
row-echelon matrix in which the leading entries on each row need not be 1. We
can achieve this using the elementary row transformations (Ri + kRj) ! (Ri)
and (Ri) ! (Rj). The ?rst transformation does not change the determinant.
The second one changes its sign. Thus we see that if U is a row-echelon form
obtain from A using row replacements or row interchanges, then
jAj = (??1)r jUj (2.3)
where r is the number of row interchanges.
Remark 155 In addition, we know that if A is invertible, then all the diago-
nal entries of U in the previous remark will be nonzero entries since A is row
equivalent to the identity matrix. Otherwise, if A is not invertible, at least one
of the diagonal entries of U will be zero, hence jAj = jUj = 0.
Combining the two remarks, we have the following proposition:
Proposition 156 If U is a row-echelon form obtain from A using row replace-
ments or row interchanges only, then assuming there are r row interchanges
performed:
jAj =

(??1)r jUj if A is invertible
0 if A is not invertible
=

(??1)r (product of the diagonal entries of U) if A is invertible
0 if A is not invertible
2.2. PROPERTIES OF DETERMINANTS 69
An immediate consequence of this result is the following important theorem.
Theorem 157 An n  n matrix A is invertible if and only if jAj 6= 0.
We ?nish this subsection with a note on the determinant of elementary
matrices.
Theorem 158 Let E be an elementary n  n matrix.
1. If E is obtained by multiplying a row of In by k, then jEj = k.
2. If E is obtained by switching two rows of In, then jEj = ??1.
3. If E is obtained by replacing a row of In by itself plus a multiple of another
row of In, then jEj = 1.
Remark 159 (Numerical Notes) Earlier, we mentioned that computing the
determinant of an nn matrix using cofactor expansion involved n! operations,
which makes it impossible for fast computers to compute even the determinant
of a 25  25 matrix (500 000 years for a machine which performs one trillion
operations per second). If we use the method outlines in the proposition, it can be
proven that it requires
2n3
3
operations. Thus, it would take
2
??
253


3
1000000000000
=
1: 041 7  10??8 seconds for a computer performing one trillion operations per
second. This is much faster.
2.2.2 Additional Properties
We begin with a few useful theorems which will make computing determinants
easier in certain cases.
Theorem 160 Let A be an nn matrix. If A has a row of zeros then jAj = 0.
Proof. The proof is straightforward. We simply do a cofactor expansion along
the row containing zeros.
Corollary 161 Let A be an n  n matrix. If A has a row which is a multiple
of another row, then jAj = 0.
Proof. Suppose that Ri = kRj . Then, if we perform an A the elementary
row operation (Ri ?? kRj) ! (Ri) and call B the resulting matrix, then the ith
row of B will consist of zeros. Since this transformation does not change the
determinant, it follows that jBj = jAj. By the theorem, jBj = 0.
Theorem 162 Let A be an n  n matrix.

AT

= jAj.
Proof. AT is obtained from A by switching its rows and columns. Since we
can compute the determinant by row or column cofactor expansion and get the
same answer, we can compute jAj by cofactor expansion along the ?rst row of
A which is the same as cofactor expansion along the ?rst column of AT . But
the latter is

AT

70 CHAPTER 2. DETERMINANTS
Remark 163 This is a very important result. Everything we said above regard-
ing rows can be restated using columns. For example a matrix with a column of
zeros has a determinant equal to 0. Similarly, a matrix for which one column
is a multiple of another has a determinant equal to 0.
Next, we look at jA + Bj, jkAj and jABj.
Theorem 164 Let A be an nn matrix and k a constant. Then jkAj = kn jAj.
Proof. This is a repeated application of theorem 151, we have

ka11 ka12


ka1n
ka21 ka22


ka2n
...
...
. . .
...
kan1 kan2


kann

= k

a11 a12


a1n
ka21 ka22


ka2n
...
...
. . .
...
kan1 kan2


kann
 =
k
2
a11 a12


a1n
a21 a22


a2n
...
...
. . .
... k
a
n
1
k
a
n
2






k
a
n
n
...
= kn
a11 a12


a1n
a21 a22


a2n
...
....
.
.
...
an1 an2


ann
Theorem 165 If A is an nn matrix and E an nn elementary matrix, then
jEAj = jEj jAj.
Proof. We consider three cases.
Case 1 E is obtained from In by interchanging two rows. On one hand, by
theorem 151, jEAj = ??jAj. But by theorem 158, jEj jAj = ??jAj. So, the
two are equal.
Case 2 E is obtained from In replacing a row by a non-zero multiple (k) of
itself. On one hand, by theorem 151, jEAj = k jAj. But by theorem 158,
jEj jAj = k jAj. So, the two are equal.
Case 3 E is obtained from In replacing one row by itself plus a multiple of
another row. On one hand, by theorem 151, jEAj = jAj. But by theorem
158, jEj jAj = jAj. So, the two are equal.
Theorem 166 If A and B are two n  n matrices, then jABj = jAj jBj.
Proof. Again, we divide the proof in two case based on the invertibility of A.
2.2. PROPERTIES OF DETERMINANTS 71
Case 1 Suppose A is not invertible. Then, AB is not invertible. Thus, by
theorem 157 we have jAj = 0 thus jAj jBj = 0 and jABj = 0.
Case 2 Suppose A is invertible. Then, A is row equivalent to In thus there exist
a sequence of elementary matrices E1;E2; :::;Ek such that A = E1E2:::Ek.
Thus,
AB = E1E2:::EkB
Hence, by repeated application of theorem 165, we have
jABj = jE1E2:::EkBj
= jE1j jE2j :: jEkj jBj
= jE1E2:::Ekj jBj
= jAj jBj
Theorem 167 If A is invertible, then

A??1

=
1
jAj
.
Proof. If A is invertible, then A??1A = I. By theorem 166, we have

A??1

jAj = jIj
= 1
hence the result.
Using what we learned in this section, we can add to theorem 116. The new
version is:
Theorem 168 If A is an nn matrix, then the following statements are equiv-
alent:
1. A is invertible.
2. Ax = b has a unique solution for any n  1 column matrix b.
3. Ax = 0 has only the trivial solution.
4. A is row equivalent to In.
5. A can be written as the product of elementary matrices.
6. jAj 6= 0
72 CHAPTER 2. DETERMINANTS
2.2.3 Introduction to Eigenvalues
Many applications in Linear Algebra involve solving an equation of the form
Ax = x where A is an n  n matrix, x is an n  1 vector, and  is a scalar.
More speci?cally, these applications seek the values of  for which the system
has nontrivial solutions.
Ax = x () Ax ?? x = 0
() (A ?? I) x = 0
From theorem 168, this system has nontrivial solutions if and only if jA ?? Ij =
0.
De?nition 169 Let A be an n  n matrix.
1. The values of  such that (A ?? I) x = 0 has nontrivial solutions are called
characteristic values or proper values or eigenvalues of the matrix
A.
2. If  is an eigenvalue of A, the corresponding nontrivial solution of the
system is called an eigenvector.
3. jA ?? Ij = 0 is called the characteristic equation of A.
Example 170 Find the eigenvalues and eigenvectors of A =

1 3
4 2

.
 Finding the Eigenvalues: We begin by writing A ?? I.
A ?? I =

1 3
4 2

?? 

1 0
0 1

=

1 ??  3
4 2 ?? 

Therefore,
jA ?? Ij =

1 ??  3
4 2 ?? 

= (1 ?? ) (2 ?? ) ?? 12
= 2 ?? 3 + 2 ?? 12
= 2 ?? 3 ?? 10
Hence, the characteristic equation is
2 ?? 3 ?? 10 = 0
Its solutions are
2 ?? 3 ?? 10 = 0 () ( ?? 5) ( + 2) = 0
()  = 5 or  = ??2
These are the eigenvalues.
2.2. PROPERTIES OF DETERMINANTS 73
 Finding the Eigenvectors: We do it for each eigenvalue.
?If  = 5, then the system becomes
(A ?? 5I) x = 0

1 ?? 5 3
4 2 ?? 5
 
x1
x2

=

0
0


??4 3
4 ??3
 
x1
x2

=

0
0

We see that the two equations are the same, therefore the solution
is 4x1 = 3x2. So, if x2 = t then x1 =
3
4
t. So, the eigenvectors
corresponding to  = 5 are the non-zero solutions of x =
"
3
4
t
t
#
?If  = ??2, then the system becomes
(A + 2I) x = 0

1 + 2 3
4 2 + 2
 
x1
x2

=

0
0


3 3
4 4
 
x1
x2

=

0
0

Once again, the two equations are the same. The solutions are x1 =
??x2. If x2 = t, then x1 = ??t. So, the eigenvectors corresponding to
 = ??2 are the non-zero solutions of x =

??t
t

2.2.4 Concepts Review
 Know how the elementary row (columns) transformations a¤ect the de-
terminant of a matrix.
 Know how to compute determinants using elementary row transforma-
tions.
 Know the relationship between the determinant of a matrix and the de-
terminant of its transpose.
 Know the relationship between the determinant of a matrix and the de-
terminant of its inverse.
 Know what eigenvalues and eigenvectors are and be able to compute them.
74 CHAPTER 2. DETERMINANTS
2.2.5 Problems
1. On pages 101, 102, do # 2, 3, 4, 5, 8, 9, 11, 12, 13, 14.
2. On pages 109 - 111, do # 4, 5, 6, 12, 13, 14, 15, 20, 22, 23.
3. Citing theorems studied, explain why a matrix with a row or column of
zeros is not invertible.
4. Citing theorems studied, explain why a matrix with a row or column which
is a multiple of another row or column is not invertible.
5. Prove that if A and B are two n  n matrices, then jABj = jBAj.