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Determinants
1. Determinants
2.
1 The Determinant of a Matrix2 Properties of Determinants
3 Application of Determinants: Cramer’s Rule
3.2
3.
1 The Determinant of a Matrix3.3
4.
Determinant  a square array ofnumbers or variables enclosed between
parallel vertical bars.
**To find a determinant you must have
a SQUARE MATRIX!!**
Finding a 2 x 2 determinant:
a
b
c d
= ad  bc
5.
Given a square matrix A its determinant is a real numberassociated with the matrix.
The determinant of A is written:
det (A)
or
A
For a 2x2 matrix, the definition is
a
b
a
b
det
=
= ad  bc
c d
c d
• For larger matrices the definition is more
complicated
6.
※ The determinant is NOT a matrix operation※ The determinant is a kind of information extracted from a
square matrix to reflect some characteristics of that square
matrix
※ For example, this chapter will discuss that matrices with a
zero determinant are with very different characteristics from
those with nonzero determinants
※ The motives to calculate determinants are to identify the
characteristics of matrices and thus facilitate the comparison
between matrices since it is impossible to investigate or
compare matrices entry by entry
※ The similar idea is to compare groups of numbers through
the calculation of averages and standard deviations
※ Not only the determinant but also the eigenvalues and
eigenvectors are the information that can be used to identify
the characteristics of square matrices
3.6
7.
The determinant of a 2 × 2 matrix:a11 a12
A
a
a
21 22
det( A)  A  a11a22
a21a12
Note:
1. For every SQUARE matrix, there is a real number
associated with this matrix and called its determinant
2. It is common practice to omit the matrix brackets
a11 a12
a11 a12
a
a 21 a 22
21 a22
3.7
8.
Historically speaking, the use of determinants arose from therecognition of special patterns that occur in the solutions of
linear systems:
a11 x1
a21 x1
a12 x2
b1
a22 x2
b2
b1a22 b2 a12
b2 a11 b1a21
x1
and x2
a11a22 a21a12
a11a22 a21a12
Note:
1. x1 and x2 have the same denominator, and this quantity is
called the determinant of the coefficient matrix A
2. There is a unique solution if a11a22 – a21a12 = A ≠ 0
3.8
9. Determinants 2x2 examples
12
det
3 4
1
2
=
3 4
= (1)(4) – (2)(3) = 2
5
2
det
2 0
5
2
=
2 0
= (5)(0) – (2)(2) = 4
1
2
det
2 4
1
2
=
2 4
= (1)(4) – (2)(2) = 0
Note: The determinant of a matrix can be positive, zero, or negative
10.
Determinants• To define det(A) for larger matrices, we will
need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix
formed by removing the ith row and the jth
column of A
1 1 2
A = 1 2 3
2 7 0
M11 : remove row 1, col 1
2
3
M11 =
7 0
11.
Determinants• To define det(A) for larger matrices, we will
need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix
formed by removing the ith row and the jth
column of A
1 1 2
A = 1 2 3
2 7 0
M12 : remove row 1, col 2
1
3
M12 =
2 0
12.
Determinants• To define det(A) for larger matrices, we will
need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix
formed by removing the ith row and the jth
column of A
1 1 2
A = 1 2 3
2 7 0
M13 : remove row 1, col 3
1
2
M13 =
2 7
13.
Determinants• To define det(A) for larger matrices, we will
need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix
formed by removing the ith row and the jth
column of A
1 1 2
A = 1 2 3
2 7 0
M21 : remove row 2, col 1
1
2
M21 =
7 0
14.
Determinants• To define det(A) for larger matrices, we will
need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix
formed by removing the ith row and the jth
column of A
1 1 2
A = 1 2 3
2 7 0
M22 : remove row 2, col 2
1
2
M22 =
2 0
15.
Determinants• To define det(A) for larger matrices, we will
need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix
formed by removing the ith row and the jth
column of A
1 1 2
A = 1 2 3
2 7 0
M23 : remove row 2, col 3
1
1
M23 =
2 7
16.
Determinants• To define det(A) for larger matrices, we will
need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix
formed by removing the ith row and the jth
column of A
1 1 2
A = 1 2 3
2 7 0
M31 : remove row 3, col 1
1
2
M31 =
2 3
17.
Determinants• To define det(A) for larger matrices, we will
need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix
formed by removing the ith row and the jth
column of A
1 1 2
A = 1 2 3
2 7 0
M32 : remove row 3, col 2
1
2
M32 =
1 3
18.
Determinants• To define det(A) for larger matrices, we will
need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix
formed by removing the ith row and the jth
column of A
1 1 2
A = 1 2 3
2 7 0
M33 : remove row 3, col 3
1
1
M33 =
1 2
19.
The formula for a 3x3 matrixFor a matrix
a11 a12 a13
A = a21 a22 a23
a31 a32 a33
• Its determinant is given by
A = a11M11  a12M12 + a13M13
• From the formula for a 2x2 matrix:
a22 a23
M11=
= a22a33  a23a32
a32 a33
20.
The formula for a 3x3 matrixFor a matrix
a11 a12 a13
A = a21 a22 a23
a31 a32 a33
• Its determinant is given by
A = a11M11  a12M12 + a13M13
• From the formula for a 2x2 matrix:
a21 a23
M12=
= a21a33  a23a31
a31 a33
21.
The formula for a 3x3 matrixFor a matrix
a11 a12 a13
A = a21 a22 a23
a31 a32 a33
• Its determinant is given by
A = a11M11  a12M12 + a13M13
• From the formula for a 2x2 matrix:
a21 a22
M13=
= a21a32  a31a22
a31 a32
22.
3x3 Example1 1 2
A = 1 2 3
2 7 0
A = 1xM11  1xM12 + (2)xM13
2
3
A= 1x
7 0
1
3
 1x
2 0
1
2
+ (2)
2 7
= 1x(21) 1x(6) +(2)x(11) = 7
23.
3x3 Example0 1 3
B= 5 3 1
1 2 0
B = 0xM11  1xM12 + 3xM13
3
1
B= 0x
2 0
5
1
 1x
1 0
= 0x(2) 1x(1) +(3)x(13) = 38
5
3
+ 3x
1 2
24.
The formula for a 3x3 matrixFor the matrix
a11 a12 a13
A = a21 a22 a23
a31 a32 a33
• We used the top row to calculate the determinant:
A = a11M11  a12M12 + a13M13
• However, we could equally have used any row of
the matrix and performed a similar calculation
25.
The formula for a 3x3 matrixFor the matrix
a11 a12 a13
A = a21 a22 a23
a31 a32 a33
• Using the top row:
A = a11M11  a12M12 + a13M13
• Using the second row
A = a21M21 + a22M22  a23M23
• Using the third row
A = a31M31  a32M32 + a33M33
26.
The formula for a 3x3 matrixA = a11M11  a12M12 + a13M13
= a21M21 + a22M22  a23M23
= a31M31  a32M32 + a33M33
• Notice the changing signs depending on what
row we use:
+  +
 + +  +
27.
The formula for a 3x3 matrix• Equally, we could have used any column as long
as we follow the signs pattern
a11 a12 a13
A = a21 a22 a23
a31 a32 a33
+  +
 + +  +
• E.g. using the first column:
A = a11M11  a21M21 + a31M31
28.
• This choice sometimes makes it a bit easier tocalculate determinants. e.g.
1 1 2
A= 0 2 3
0 1 1
• Using the first row:
2
3
A= 1x
1 1
0
3
 1x
0 1
0
2
+ (2) x
0 1
= 1x(1) 1x(0) + (2)x(0) = 1
29.
• This choice sometimes makes it a bit easier tocalculate determinants. e.g.
1 1 2
A= 0 2 3
0 1 1
• However, using the first column:
2
3
A= 1x
1 1
 0 + 0 = 1x(1) = 1
30.
A general formula for determinants• For a 4x4 matrix we add up minors like the 3x3
case, and again use the same signs pattern
+
+

+
+
+
+

+
+
• Notice that if we think of the signs pattern as a
matrix, then it can be written as (1)i+j
31.
A general formula for determinants32.
3.3233.
Minor of the entry aij: the determinant of the matrix obtainedby deleting the ith row and jth column of A
a11
a12
M ij
a1( j 1)
a1( j 1)
a1n
a( i 1)1
a( i 1)( j 1)
a( i 1)( j 1) a( i 1) n
a( i 1)1
a( i 1)( j 1)
a( i 1)( j 1) a( i 1) n
a n1
an ( j 1)
an ( j 1)
ann
Aij ( 1) i j M ij
3.33
34.
Ex:a11
A a21
a31
M 21
a32
a13
a23
a33
a12
a13
a32
a33
a12
a22
A21 ( 1) 2 1 M 21 M 21
M 22
a13
a31
a33
А22 ( 1) 2 2 M 22 M 22
Notes: Sign pattern for cofactors. Odd
positions (where i+j is odd) have negative
signs, and even positions (where i+j is even)
have positive signs. (Positive and negative
signs appear alternately at neighboring
positions.)
a11
3.34
35.
Theorem: Expansion by cofactorsLet A be a square matrix of order n, then the determinant of A
is given by
n
(a) det( A)  A  aij Aij ai1 Ai1 ai 2 Ai 2
j 1
ain Ain
(cofactor expansion along the ith row, i=1, 2,…, n)
or
n
(b) det( A)  A  aij Aij a1 j A1 j a2 j A2 j
i 1
anj Anj
(cofactor expansion along the jth column, j=1, 2,…, n)
※The determinant can be derived by performing the cofactor expansion
along any row or column of the examined matrix
3.35
36.
Ex: The determinant of a square matrix of order 3a11
A a21
a31
a12
a22
a32
a13
a23
a33
det( A) a11 A11 a12 A12 a13 A13 (first row expansion)
a21 A21 a22 A22 a23 A23 (second row expansion)
a31 A31 a32 A32 a33 A33 (third row expansion)
a11 A11 a21 A21 a31 A31 (first column expansion)
a12 A12 a22 A22 a32 A32 (second column expansion)
a13 A13 a23 A23 a33 A33 (third column expansion)
3.36
37.
Ex: The determinant of a square matrix of order 30 2 1
A 3 1 2
4 0 1
Sol:
1 1
A11 ( 1)
1 2
0
1 3
A13 ( 1)
1
det( A) ?
1
3 1
4
0
1 2
A12 ( 1)
3 2
4 1
( 1)( 5) 5
4
det( A) a11 A11 a12 A12 a13 A13
(0)( 1) (2)(5) (1)(4)
14
3.37
38.
Ex: The determinant of a square matrix of order 41 2
1 1
A
0
2
4
3
0
0 2
det( A) ?
0 3
0 2
3
3.38
39.
Sol:det( A) (3)( A13 ) (0)( A23 ) (0)( A33 ) (0)( A43 ) 3 A13
1 1
3( 1)1 3 0
3
2
2
3
4 2
2
2
2 1 1
2 2 1
2 3 1 1
3 (0)( 1)
( 2)( 1)
(3)( 1)
4
2
3
2
3
4
3 0 (2)(1)( 4) (3)( 1)( 7)
(3)(13)
39
※ By comparing the exercises, it is apparent that the computational effort for the
determinant of 4×4 matrices is much higher than that of 3×3 matrices.
3.39
40.
Upper triangular matrix:All entries below the main diagonal are zeros
Lower triangular matrix:
All entries above the main diagonal are zeros
Diagonal matrix:
All entries above and below the main diagonal are zeros
Ex:
a11 a12 a13
0 a22 a23
0 0 a
33
a11 0 0
a21 a22 0
a a a
31 32 33
a11 0 0
0 a22 0
0 0 a
33
upper triangular
lower triangular
diagonal
3.40
41.
Theorem: (Determinant of a Triangular Matrix)If A is an n n triangular matrix (upper triangular, lower
triangular, or diagonal), then its determinant is the product of
the entries on the main diagonal. That is
det( A)  A  a11a22 a33 ann
3.41
42. Ex: Find the determinants of the following triangular matrices
02
4 2
(a) A
5 6
1
5
0 0
0 0
1 0
3 3
Sol:
1
0
(b) B 0
0
0
0 0 0
3 0 0
0 2 0
0 0 4
0 0 0
0
0
0
0
2
(a) A = (2)(–2)(1)(3) = –12
(b) B = (–1)(3)(2)(4)(–2) = 48
3.42
43. 2 Properties of Determinants
Conditions that yield a zero determinantIf A is a square matrix and any of the following conditions is
true, then det(A) = 0
(a) An entire row (or an entire column) consists of zeros
(b) Two rows (or two columns) are equal
(c) One row (or column) is a multiple of another row (or column)
3.43
44.
Ex:1 2 3
1 4 0
1 1 1
0 0 0 0
2 5 0 0
2 2 2 0
4 5 6
3 6 0
4 5 6
1 4 2
1
2
3
1
1 5 2 0
4
5
6 0
2 10 5 0
1 6 2
2 4 6
8
4
3 12 6
3.44
45.
Theorem: Determinant of a matrix productdet(AB) = det(A) det(B)
Notes:
(1) det( A1 A2
(Verified by Ex on the next slide)
(2)
An ) det( A1 ) det( A2 )
det( An )
det( A B) det( A) det( B)
a11
a12
a13
a11
a12
a13
a11
a12
a13
(3) a21 b21 a22 b22 a23 b23 a21 a22 a23 b21 b22 b23
a31
a32
a33
a31
a32
a33
a31
a32
a33
3.45
46.
Ex 1: The determinant of a matrix product1 2 2
A 0 3 2
1 0 1
1
2 0
B 0 1 2
3 1 2
Find A, B, and AB
Sol:
1 2 2
 A  0
3
2 7
1
0
1
2
0
1
 B  0 1 2 11
3
1
2
3.46
47.
1 8 41
1 2 2 2 0
AB 0 3 2 0 1 2 6 1 10
1 0 1 3 1 2 5 1
1
8
4
1
 AB  6 1 10 77
5
1
1
Check:
AB = A B
3.47
48.
Theorem: Determinant of a scalar multiple of a matrixIf A is an n × n matrix and c is a scalar, then
det(cA) = cn det(A)
Ex 2:
(can be proven by repeatedly use the fact
that
)
if B M i( k ) ( A) B k A
1 2 4
10 20 40
A 30
0 50 , if 3 0 5 5, find A
20 30 10
2 3 1
Sol:
1 2 4
1 2 4
3
0 5 (1000)(5) 5000
A 10 3
0 5 A 10 3
2 3 1
2 3 1
3.48
49.
Theorem: (Determinant of an invertible matrix)A square matrix A is invertible (nonsingular) if and
only if det(A) 0
3.49
50.
Ex 3: Classifying square matrices as singular or nonsingular0 2 1
A 3 2 1
3 2 1
0 2 1
B 3 2 1
3 2
1
Sol:
A 0
A has no inverse (it is singular)
B 12 0
B has inverse (it is nonsingular)
3.50
51. Inverse Matrices
3.5152. Theorem of Inverse Matrices
3.5253.
3.5354.
3.5455.
3.5556. Example 3
3.5657.
3.5758.
Theorem: Determinant of an inverse matrix1
If A is invertible, then det( A )
det( A)
1
Theorem: Determinant of a transpose
If A is a square matrix, then det( AT ) det(A)
Ex 4:
Sol:
1 0 3
A 0 1 2
2 1 0
1
0
3
 A  0 1 2 4
2
1
0
1
(a) A ?
(b)
AT ?
1 1
A 4
AT A 4
A 1
3.58
59.
The similarity between the noninvertible matrix and the realnumber 0
Matrix A
Invertible
det( A) 0
A 1 exists and det( A 1 )
Noninvertible
Real number c
c 0
1
det( A)
c 1 exists and c 1 =
det( A) 0
c 0
A 1 does not exist
c 1 does not exist
1
1
1
det( A )
det(
A
)
0
1 1 1
c = =
c 0
1
c
3.59
60.
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 matrix b
(3) Ax = 0 has only the trivial solution
(4) det(A) 0
3.60
61.
Ex 5: Which of the following system has a unique solution?2 x2
x3
1
3x1
2 x2
x3
4
3x1
2 x2
x3
4
2 x2
x3
1
3x1
2 x2
x3
4
3x1
2 x2
x3
4
(a)
(b)
3.61
62.
Sol:(a) Ax b (the coefficient matrix is the matrix A in Ex 3)
A 0 (from Ex 3)
This system does not have a unique solution
(b) Bx b (the coefficient matrix is the matrix B in Ex 3)
B 12 0 (from Ex 3)
This system has a unique solution
3.62
63. 3 Applications of Determinants
Theorem: Cramer’s Rulea11 x1 a12 x2
a1n xn b1
a21 x1 a22 x2
a2 n xn b2
an1 x1 an 2 x2
ann xn bn
where A aij
n n
A(1) A(2)
Ax b
A(i) represents the ith
column vector in A
b1
x1
b
x
2
2
( n)
b
x
,
A ,
x
bn
n
Suppose this system has a unique solution, i.e.,
det( A)
a11
a12
a1n
a21
a22
a2 n
an1
an 2
ann
0
3.63
64.
By defining Aj A(1) A(2)a11
a
21
an1
A( j 1) b A( j 1)
a1( j 1)
b1
a1( j 1)
a2( j 1)
b2
a2( j 1)
an ( j 1)
bn
an ( j 1)
(i.e., det( Aj ) b1C1 j b2C2 j
xj
det( Aj )
det( A)
, j 1, 2,
A( n )
a1n
a2 n
ann
bnCnj )
,n
3.64
65.
Ex: Use Cramer’s rule to solve the system of linear equationx 2 y 3z 1
2x
z 0
3x 4 y 4 z 2
Sol:
1 2 3
1 2 3
det( A1 ) 0 0
1 8
det( A) 2
0
1 10
2 4 4
3 4 4
1
1 1 3
det( A2 ) 2
0
1 15,
3
2
4
det( A1 ) 4
x
det( A) 5
det( A3 ) 2
det( A2 ) 3
y
det( A)
2
3
2
1
0
0 16
4 2
det( A3 ) 8
z
det( A)
5
3.65
66. Keywords
determinantminor
cofactor
expansion by cofactors
upper triangular matrix
lower triangular matrix
diagonal matrix
Cramer’s rule
3.66