Chap. 3
Determinants
3.1
3.2
3.3
3.4
3.5
The Determinants of a Matrix
Evaluation of a Determinant Using Elementary Operations
Properties of Determinants
Introduction to Eigenvalues
Applications of Determinants
3.1 The Determinant of a Matrix



Every square matrix can be associated with a real number
called its determinant.
 a11 a12 
Definition: The determinant of the matrix A  

a
a

 21 22 
a11 a12
is given by det( A)  A 
 a11a22  a21a12
a21 a22
+
2 3
Example 1:
 ?2(2)  1(3) = 7 A  [2]  A  ?2
1 2
2 1
 ?2(2)  1(4) = 0
4 2
0 3
Ming-Feng Yeh
2 4
 ?0(4)  2(3) = 6
Chapter 3
3-2
Section 3-1
Minors and Cofactors of a Matrix



If A is a square matrix, then the minor (子行列式) Mij of
the element aij is the determinant of the matrix obtained
by deleting the ith row and jth column of A.
The cofactor (餘因子) Cij is given by Cij = (1)i+jMij.
 a11 a12
A  a21 a22
a31 a32
a13  M  a12 a13  C  (1) 21 M  M
21
21
21
21

a
a
32
33
a23 
a
a
a33  M 22  11 13  C22  (1) 22 M 22  M 22
a31 a33
Sign pattern for cofactors:
Ming-Feng Yeh
   
  


    33
Chapter 3






  
   
  

    44
3-3
Section 3-1
Theorem 3.1
Expansion by Cofactors
 Let A be a square matrix of order n. Then the determinant
of A is given by
n
ith row
det( A)  A   aijCij  ai1Ci1  ai 2Ci 2    ainCin expansion
j 1
n
column
det( A)  A   aijCij  a1 j C1 j  a2 j C2 j    anjCnj jth
expansion
i 1

For any 33 matrix:
a11 a12
a21 a22
a31 a32


a13 a11 a12
a23 a21 a22
a33 a31 a32

 a11 a12 a13 
A  a21 a22 a23 
a31 a32 a33 
+
+
+
A  a11a22a33  a12a23a31  a13a21a32  a31a22a13  a32a23a11  a33a21a12
Ming-Feng Yeh
Chapter 3
3-4
Section 3-1
0 2 1
A   3  1 2
4 0 1
Examples 2 & 3
Find all the minors and cofactors of A, and then find the
determinant of A.
1 2
Sol:
3 2
3 1

M 11 
0
1
 1, M 12 
4
1
 5,
M 13 
4
0
4
C13  4
C11  1
C12  5
M 21  2, M 22  4, M 23  8, C21  2, C22  4, C23  8,
M 31  5, M 32  3, M 33  6. C31  5, C32  3, C33  6.
A  a11C11  a12C12  a13C13  0(1)  2(5)  1(4)  14
 a21C21  a22C22  a23C23  3(2)  (1)( 4)  2(8)  14
 a11C11  a21C21  a31C31  0(1)  3(2)  4(5)  14
Ming-Feng Yeh
Chapter 3
3-5
Section 3-1
Example 5
2 1
0



1
2
 Find the determinant of A  3


Sol:
(4) (0) (6) 4  4 1
2
0
2 1 0
0
2 1
3 1 2 3 1
3 1 2
4 4 1 4 4
4 4 1
+(0) +(16) +(12)
A  0  16  (12)  (4)  0  6  2
Ming-Feng Yeh
Chapter 3
3-6
Section 3-1
Example 4
0
 1 2 3
 1

1
0
2

 Find the determinant of A  
 0
2 0
3
Sol: Expansion by which row 

3
4
0

2


or which column?
 the 3rd column: three of the entires are zeros
1 1
1 1
2  (1)( 2)( 2)  0(4)( 2)  1(3)(3)
C13  (1)13 0 2
3 0 2
3
 3(2)( 2)  4(3)( 1)  0(1)( 2)
3 4 2
3 4  2  4  9  12  12  13
2
2
1
2 1 1
2 2  1
23  1
 (0)( 1)
 (2)( 1)
 (3)( 1)
4 2
3 2
3 4
2
 0  (2)( 4)  (3)( 7)  13
Ming-Feng Yeh
A  a13C13  3(13)  39
Chapter 3
3-7
Section 3-1
Triangular Matrices
Upper triangular Matrix
Lower triangular Matrix
a11 a12 a13 
 0 a
a23 
22

 0
0 a33 


 
 
 0
0
0 
0
0
 a11
a
0
 21 a22
 a31 a32 a33



 
 an1 an 2 an 3

a1n 
a2 n 
a3n 


ann 

0

0

0



 ann 
Theorem 3.2: If A is a triangular matrix of order n, then its
determinant is the product of the entires on the main
diagonal. That is, det( A)  A  a11a22a33 ann
Ming-Feng Yeh
Chapter 3
3-8
Section 3-1
Example
2 3 1
11  1 2
 2[( 1)(3)  0(2)]  6
0  1 2  ?2(1)
0 3
0 0 3
1 0 0 0
0
2
0 0 0
0 3 0 0
0
4 2 0 0
?
0 0 2 0
0 ?
5
6 1 0
0 0 0 4
0
1
5 3 3
0 0 0 0 2
 2(2)(1)(3)
 (1)(3)( 2)( 4)( 2)
 12
 48
Ming-Feng Yeh
Chapter 3
3-9
3.2 Evaluation of a Determinant
Using Elementary Operations
Which of the following two determinants is easier to evaluate?
1 2
3
1
6
3
2
4
3
2
4 6
3
2  1 4  9  3  (2)( 1)  2  9  3
A
2
4 9 3
6
9
2
3
9
2
3 6
9
2
4
6
2
4 6
3

By elementary row operations
1 2
B
3 2
4  3  1(1)  2
4 9
3 6
2
3 6
9
0
1  1(60)  2(39)  3(10)  (18)  6
2 9 2
0  3  1  (1)( 2)( 3)( 1)  6
0
0
0
Ming-Feng Yeh
3
0
1
Chapter 3
3-10
Section 3-2
 Theorem 3.3
Elementary Row Operations and Determinants
 Let A and B be square matrices.
1. If B is obtained from A by interchanging two rows of A,
then det(B) = det(A).
2. If B is obtained from A by adding a multiple of a row of A
to another row of A, then det(B) = det(A).
3. If B is obtained from A by multiplying a row of A by a
nonzero constant c, then det(B) = cdet(A).
Take a common factor out of a row
2 1 3
A
2
4 3
(2)
Ming-Feng Yeh
4 3
1. B 
 2
2 1
2 1
2. B 
2
0 1
Chapter 3
6 3
3. B 
6
4 3
3-11
Section 3-2
Example 2
2  3 10


2

2
 Find the determinant of A  1


0
Sol:
1  3
1
2 2
1
2  2 (2)
2  3 10
2  2   2  3 10
0
1 3
1 3
1
0
1 2 2
 70 1 2
0
Ming-Feng Yeh
 0 7
0
14
1 3
Factor 7 out of the 2nd row
1 2 2
 7 0 1  2  7(1)(1)( 1)  7
(1)
1 3
0 0 1
Chapter 3
3-12
Section 3-2
Determinants and
Elementary Column Operations

Although Theorem 3.3 was stated in terms of elementary
row operations, the theorem remains valid if the word “row”
is replaced by the word “column.”

Operations performed on the column of a matrix are called
elementary column operations.

Two matrices are called column-equivalent if one can be
obtained from the other by elementary column operations.
Ming-Feng Yeh
Chapter 3
3-13
Section 3-2
Example 3
2
2
 1
 3 6

 Find the determinant of A 
4


Sol:
 5  10  3
1
3
2
1 0
2
4  3 0
4  (0)C12  (0)C22  (0)C32  0
5  10  3
5 0 3
2
6
(2)
Expansion by the
second column
Ming-Feng Yeh
Chapter 3
3-14
Section 3-2
 Theorem 3.4
Conditions That Yield a Zero Determinant
 If A is a square matrix and any one of the following
conditions is true, then det(A) = 0.
1. An entire row (or an entire column) consists of zeros.
2. Two rows (or columns) are equal.
3. One row (or column) is a multiple of another row (or
(3)
column).
0
2
0
0
4 5  0
3 5
Ming-Feng Yeh
2
(1) 1  2 4
0
1 2 0
1 2 4
Chapter 3
1 2 3
2 1  6  0
2
0
6
3-15
Section 3-2
Examples 4 & 5
(2)

1 4 1 1
4
1
2  1 0  ?0  9  2  0
0 18 4 0 18
4

3
5 2 3
5 4
5 4
31
2  4 1  ? 2  4
3  3(1)
4
3
3
0 6 3
0
0
(2)
Ming-Feng Yeh
 3(1)( 1)  3
Chapter 3
3-16
Section 3-2
Example 6
1
 2 0
 2
1 3

0 1
 Find the determinant of A   1

Sol:
 3 1 2
2 0
1 3 2
 1 1 3
2 1 3 2
A  1 0 1 2
1 0
3 0
1
3
2
1
2 2 1  1
5 6  4  (1)( 1) 1 5
0 0
1
3 0
8
3  2
2  1
2
3

4  3
2
0
3 2
8
1
2
3
 8 1

6 4
13 5
0
1
0 0
(3)
(1)
3 2
2
3
6 4
0
1
1 3
0 0 5
 8 1
 (1)( 1) 4 4  8  1 2   8  1 2  5(1) 2 2
 5(40  13)
13 5
13 5 6
 135
13 5 6
Ming-Feng Yeh
Chapter 3
3-17
3.3 Properties of Determinants
Example 1: Find A , B , and AB for the matrices
1
 1  2 2
2 0
A  0
3 2 and B  0  1  2
 1
 3 1  2
0 1
Sol: 1  2 2
2 0
1

A0
1
3 2  7 and
0 1
B  0  1  2  11  A B  77
3
1 2
1 8 4
1
 1  2 2  2 0
AB  0
3 2 0  1  2  6  1  10  AB  77
 1
0 1  3 1  2 5
1  1
Ming-Feng Yeh
Chapter 3
3-18
Section 3-3
Theorems 3.5 & 3.6
Theorem 3.5: Determinant of a Matrix Product
 If A and B are square matrices of order n, then
det(AB) = det(A) det(B)
Remark: A1 A2 A3  Ak  A1 A2 A3  Ak
A  B  A B
Theorem 3.6: Determinant of a Scalar Multiple of a Matrix
 If A is a nn matrix and c is a scalar, then the determinant
of cA is given by det(cA) = cn det(A).
Remark: [Thm. 3.3] If B is obtained from A by multiplying a
row of A by a nonzero constant c, then det(B) = cdet(A).
Ming-Feng Yeh
Chapter 3
3-19
Section 3-3
Example 2
 10  20 40
 30

0
50
 Find the determinant of the matrix A 


Sol:  1  2 4
 20  30 10
1 2 4
A  10 3
0 5  3
0 5 5
 2  3 1
2 3 1
1 2 4
 A  103 3
0 5  1000(5)  5000
2 3 1
 6 2
 A
2 1  A  2
 9 9


A B  
 A  B  18

 3 7
 2 0
 AB
B
 B  3

0  1
Ming-Feng Yeh
Chapter 3
3-20
Section 3-3
 Theorems 3.7 & 3.8
Theorem 3.7: Determinant of an Invertible Matrix
 A square matrix A is invertible (nonsingular) if and only if
det(A)  0.
Theorem 3.8: Determinant of an Inverse Matrix
1
 If A is invertible, then det(A ) = 1 / det(A).
Hint: A is invertible
 AA1 = I
 AA1  I  1
Ming-Feng Yeh
Chapter 3
3-21
Section 3-3
Example 3 & 4
Example 3: Which of the matrices has an inverse?
2  1
2  1
0
0
A   3  2
1
B   3  2
1
 3
 3
2  1
2
1
Sol: A  0 (singular)
B  12  0 (nonsingul ar)
It has an inverse.
It has no inverse.
1
Example 4: Find A for the matrix
Sol:
 1 0 3
1 1
1

A


A 4


A   3  1 2
A 4
2
1 0
Ming-Feng Yeh
Chapter 3
3-22
Section 3-3
 Equivalent Conditions for a
Nonsingular Matrix
If A is an nn matrix, then the following statements are
equivalent.
1. A is invertible.
2. Ax = b has a unique solution for every n1 column vector b.
3. Ax = O has only the trivial solution.
4. A is row-equivalent to In.
5. A can be written as the product of elementary matrices.
【  Also see in Theorem 2.15 】
6. det(A)  0.
【 See Example 5 (p.148) for instance 】

Ming-Feng Yeh
Chapter 3
3-23
Section 3-3
Determinant of a Transpose
Theorem 3.9: If A is a square matrix, then det(A)=det(AT).
Example 6: Show that A  AT for the following matrix.
1  2 pf:
 3
A   2 0
0 A  (2)( 1) 21 1  2  (2)(3)  6
1
5
 4  1

5
 3 2  4
1 1
T
1 2


T

A

(
2
)(

1
)
 (2)(3)  6
A   1 0  1
2 5
 2 0
5
Thus, A  AT .
Ming-Feng Yeh
Chapter 3
3-24
3.4 Introduction to Eigenvalues
See Chapter 7
Ming-Feng Yeh
Chapter 3
3-25
3.5 Applications of Determinants

The Adjoint of a Matrix
If A is a square matrix, then the matrix of cofactors of A
C11 C12
C
C22
21

has the form
 


Cn1 Cn 2

 C1n 
 C2 n 
  

 Cnn 
The transpose of this matrix
C11 C21  Cn1 
is called the adjoint of A and
C

C

C
22
n2 
is denoted by adj(A).
adj( A)   12
 

C1n
Ming-Feng Yeh
Chapter 3
   

C2 n  Cnn 
3-26
Section 3-5
Example 1
3
2
 1
 0 2

1
 Find the adjoint of A 


Sol:
 1
0  2
The matrix of cofactors of A:
2
1
0
1
0 2



0

2
1

2
1
0


2 1
2
1 3 
 3



0

2
1

2
1
0


3 2
1 2 1
3

 2 1  0 1

0

2


 4 1 2
 6 0 3
7 1 2
Ming-Feng Yeh
1
3
2
0 2
1
1
1
1
3
0 2
1
0 2
3
2 1
0 2
4 6 7 
 adj( A)   1 0 1
2 3 2
Chapter 3 
1
0
3
0 2
3-27
Section 3-5
 Theorem 3.10
The Inverse of a Matrix Given by Its Adjoint
1
 If A is an nn invertible matrix, then A 

1
adj( A)
det( A)
a b 
If A is 22 matrix A  
,

c d 
 d  b
then the adjoint of A is adj( A)  
.

a
 c
Form Theorem 3.10 you have
1
1  d  b
1
A  adj( A) 
a 
A
ad  bc  c
Ming-Feng Yeh
Chapter 3
3-28
Section 3-5
Example 2
3
2
 1
 0 2
 to find A1 .
1
 Use the adjoint of A 


 1
0  2
Sol: A  (1)(2)(2)  (3)(1)(1)  (1)(2)(2)  3
4 6 7 
adj( A)   1 0 1
2 3 2
4 6 7  43
1
1
A1 
adj( A)   1 0 1   13
A
3
2 3 2  23
Check AA1  I ?
Ming-Feng Yeh
Chapter 3
2
0
1
7
3
1
3
2
3




3-29
Section 3-5
Theorem 3.11: Cramer’s Rule


If a system of n linear equations in n variables has a
coefficient matrix with a nonzero determinant A ,
then the solution of the system is given by
det( An )
det( A1 )
det( A2 )
x1 
, x2 
, , xn 
,
det( A)
det( A)
det( A)
where the ith column of Ai is the column of constants in
the system of equations.
a11 a12 b1
a11x1  a12 x2  a13 x3  b1
A3

a
x

a
x

a
x

b

x

 a21 a22 b2
 21 1 22 2 23 3
2
3
A
a x  a x  a x  b
a31 a32 b3
 31 1 32 2 33 3 3
Ming-Feng Yeh
Chapter 3
a11
a12
a13
a21 a22
a31 a32
a23
a33
3-30
Section 3-5
Example 4

Use Cramer’s Rule to solve the system of linear equation
 x  2 y  3z  1
for x. 2 x
 z 0
Sol:
3x  4 y  4 z  2
1
2 3
A 2
0
1  10  0 (the system has an unique solution)
3 4
4
3
8
1
2 3
y , z
2
5
0
0
1
A1
2 4
4 (2)(1)( 2)  (4)(1)(1) 8 4
x




A
10
10
10 5
Ming-Feng Yeh
Chapter 3
3-31
Section 3-5
Area of a Triangle

The area of a triangle whose vertices
are (x1, y1), (x2, y2), and (x3, y3) is
x1 y1 1
1
given by Area   x2 y2 1
2
x3 y3 1
where the sign () is chosen to give a positive area.
1
pf: Area = 12 ( y1  y3 )( x3  x1 )  2 ( y3  y2 )( x2  x3 ) 2 ( y1  y2 )( x2  x1 )
1
 12 ( x1 y2  x2 y3  x3 y1  x1 y3  x2 y1  x3 y2 )
x1 y1 1
1
 x2 y 2 1
2
x3 y3 1
Ming-Feng Yeh
Chapter 3
3-32
Section 3-5
Example 5
Fine the area of the triangle whose vertices are (1, 0), (2, 2),
and (4, 3).
(4,3)
Sol: 1 0 1
1
3
3
2 2 1    Area 
2
2
2
4 3 1

Fine the area of the triangle whose
vertices are (0, 1), (2, 2), and (4, 3).
(2,2)
(1,0)
0 1 1
1
2 2 1  0  Area  0
2
4 3 1
Three points in the xy-plane lie on the same line.
Ming-Feng Yeh
Chapter 3
3-33
Section 3-5
Collinear Pts & Line Equation


Test for Collinear Points in the xy-Plane
Three points (x1, y1), (x2, y2), and (x3, y3) are collinear
x1 y1 1
if and only if x2 y2 1  0
x3 y3 1
Two-Point Form of the Equation of a Line
An equation of the line passing through the distinct points
x y 1
(x1, y1) and (x2, y2) is given by x1 y1 1  0
x2 y 2 1
The 3rd point: (x, y)
Ming-Feng Yeh
Chapter 3
3-34
Section 3-5
Example 6
Find an equation of the line passing through the points
(2, 4) and (1, 3).
Sol: x y 1

2 4 1 0
1 3 1
2 1
2 4
4 1

y

1
0
x
1 1
1 3
3 1
 x  3 y  10  0
An equation of the line is x  3y = 10.
Ming-Feng Yeh
Chapter 3
3-35
Section 3-5
Volume of Tetrahedron

The volume of the tetrahedron
whose vertices are (x1,y1, z1), (x2,
y2, z2), (x3, y3, z3), and (x4, y4, z4),
is given by
x y z 1
1
1
1 x2
Volume  
6 x3
x4
y2
y3
y4
1
z2 1
z3 1
z4 1
where the sign () is chosen to
give a positive area.
Ming-Feng Yeh
Chapter 3
Example 7: Find the volume of
the tetrahedron whose vertices
are (0,4,1), (4,0,0), (3,5,2), and
(2,2,5).
Sol: 0 4 1 1

14 0 0 1 1
 (72)  12
63 5 2 1 6
2 2 5 1
 Volume  12
3-36
Section 3-5
Coplanar Pts & Plane Equation

Test for Coplanar Points
in Space
Four points (x1,y1, z1), (x2,
y2, z2), (x3, y3, z3), and (x4,
y4, z4) are coplanar if and
only if
x1 y1 z1 1
x2
x3
x4
Ming-Feng Yeh
y2
y3
y4
z2 1
0
z3 1
z4 1

Three-Point Form of the
Equation of a Plane
An equation of the plane
passing through the
distinct points (x1,y1, z1),
(x2, y2, z2), and (x3, y3, z3)
is given by
x y z 1
x1
x2
x3
Chapter 3
y1
y2
y3
z1 1
0
z2 1
z3 1
3-37
Section 3-5
Example 8
Find an equation of the plane passing through the points
(0,1,0), (1,3,2) and (2,0,1).
(1)
Sol:

x
x y z 1
0
0 1 0 1
0 
1
1 3 2 1
2
2 0 1 1
x
 1
2
Ming-Feng Yeh
y 1 z 1
0
0 1
0
2
2 1
1
1 1
y 1 z
2
1
2 0
1
 4 x  3 y  5 z  3
Chapter 3
3-38