chapter 3 determinants 3.1 the determinant of a matrix 3.2 evaluation of a determinant using...
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Chapter 3
Determinants
3.1 The Determinant of a Matrix
3.2 Evaluation of a Determinant using Elementary Row Operations
3.3 Properties of Determinants
3.4 Application of Determinants
3.1
3.2
※ 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 zero determinant are with very different characteristics from those with non-zero determinant ※ The motive to find this information is to identify the characteristics of matrices and thus facilitate the comparison between matrices since it is impossible to 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
3.3
3.1 The Determinant of a Matrix
The determinant (行列式 ) of a 2 × 2 matrix:
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
2221
1211
aa
aaA
12212211||)det( aaaaAA
2221
1211
aa
aa
2221
1211
aa
aa
3.4
Historically speaking, the use of determinants arose from the recognition of special patterns that occur in the solutions of linear systems:
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
11 1 12 2 1
21 1 22 2 2
1 22 2 12 2 11 1 211 2
11 22 21 12 11 22 21 12
and
a x a x b
a x a x b
b a b a b a b ax x
a a a a a a a a
3.5
Ex. 1: The determinant of a matrix of order 2
734)3(1)2(221
32
044)1(4)2(224
12
330)2/3(2)4(042
2/30
Note: The determinant of a matrix can be positive, zero, or negative
3.6
Minor (子行列式 ) of the entry aij: the determinant of the matrix obtained by deleting the i-th row and j-th column of A
Cofactor (餘因子 ) of aij:
ijji
ij MC )1(
nnjnjnn
nijijii
nijijii
njj
ij
aaaa
aaaa
aaaa
aaaaa
M
)1()1(1
)1()1)(1()1)(1(1)1(
)1()1)(1()1)(1(1)1(
1)1(1)1(11211
※ Mij is a real number
※ Cij is also a real number
3.7
Ex:
333231
232221
131211
aaa
aaa
aaa
A
3332
131221 aa
aaM
212112
21 )1( MMC
3331
131122 aa
aaM
222222
22 )1( MMC
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.)
3.8
Theorem 3.1: Expansion by cofactors (餘因子展開 )
1 1 2 21
(a) det( ) | |n
ij ij i i i i in inj
A A a C a C a C a C
(cofactor expansion along the i-th row, i=1, 2,…, n)
1 1 2 21
(b) det( ) | |n
ij ij j j j j nj nji
A A a C a C a C a C
(cofactor expansion along the j-th column, j=1, 2,…, n)
Let A be a square matrix of order n, then the determinant of A
is given by
or
※The determinant can be derived by performing the cofactor expansion along any row or column of the examined matrix
3.9
Ex: The determinant of a square matrix of order 3
333231
232221
131211
aaa
aaa
aaa
A
11 11 12 12 13 13
21 21 22 22 23 23
31 31 32 32 33 33
11 11 21 21 31 31
12 12 22 22 3
det( ) (first row expansion)
(second row expansion)
(third row expansion)
(first column expansion)
A a C a C a C
a C a C a C
a C a C a C
a C a C a C
a C a C a
2 32
13 13 23 23 33 33
(second column expansion)
(third column expansion)
C
a C a C a C
3.10
Ex 3: The determinant of a square matrix of order 3
?)det( A
0 2 1
3 1 2
4 0 1
A
110
21)1( 11
11
C
Sol:
5)5)(1(14
23)1( 21
12 C
404
13)1( 31
13
C
14
)4)(1()5)(2()1)(0(
)det( 131312121111
CaCaCaA
3.11
Alternative way to calculate the determinant of a square matrix of order 3:
11 12 13
21 22 23
31 32 33
a a a
A a a a
a a a
3231333231
2221232221
1211131211
aaaaa
aaaaa
aaaaa
122133112332
132231322113312312332211
||)det(
aaaaaa
aaaaaaaaaaaaAA
Add these three products
Subtract these three products
3.12
Ex: Recalculate the determinant of the square matrix A in Ex 3
0 2 1
3 1 2
4 0 1
A
0 2
3 1
4 0
–4
0
det( ) | | 0 16 0 ( 4) 0 6 14A A
16 0
0 6
※ This method is only valid for matrices with the order of 3
3.13
Ex 4: The determinant of a square matrix of order 4
2043
3020
2011
0321
A ?)det( A
3.14
Sol:))(0())(0())(0())(3()det( 43332313 CCCCA
243
320
211
)1(3 31
133C
39
)13)(3(
)7)(1)(3()4)(1)(2(03
43
11)1)(3(
23
21)1)(2(
24
21)1)(0(3 322212
※ By comparing Ex 4 with Ex 3, it is apparent that the computational effort for the determinant of 4×4 matrices is much higher than that of 3×3 matrices. In the next section, we will learn a more efficient way to calculate the determinant
3.15
Upper triangular matrix (上三角矩陣 ):
Lower triangular matrix (下三角矩陣 ):
Diagonal matrix (對角矩陣 ):
All entries below the main diagonal are zeros
All entries above the main diagonal are zeros
All entries above and below the main diagonal are zeros
33
2322
131211
000
aaaaaa
333231
2221
11
000
aaaaa
a
Ex:
upper triangular
33
22
11
000000
aa
a
lower triangular diagonal
3.16
Theorem 3.2: (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
nnaaaaAA 332211||)det(
※ On the next slide, I only take the case of upper triangular matrices for example to prove Theorem 3.2. It is straightforward to apply the following proof for the cases of lower triangular and diagonal matrices
3.17
Pf: by Mathematical Induction (數學歸納法 )
Suppose that the theorem is true for any upper triangular
matrix U of order n–1, i.e.,
Then consider the determinant of an upper triangular matrix A
of order n by the cofactor expansion across the n-th row
21 2 ( 1)| | 0 0 0 ( 1) n
n n n n nn nn nn nn nn nnA C C C a C a M a M
Since Mnn is the determinant of a (n–1)×(n–1) upper triangular
matrix by deleting the n-th row and n-th column of A, we can
apply the induction assumption to write
11 22 ( 1)( 1) 11 22 ( 1)( 1)| | ( )nn nn nn n n n n nnA a M a a a a a a a a
11 22 33 ( 1)( 1)| | n nU a a a a
3.18
Ex 6: Find the determinants of the following triangular matrices
(a)
3351
0165
0024
0002
A (b)
20000
04000
00200
00030
00001
B
|A| = (2)(–2)(1)(3) = –12
|B| = (–1)(3)(2)(4)(–2) = 48
(a)
(b)
Sol:
3.19
Keywords in Section 3.1:
determinant: 行列式 minor: 子行列式 cofactor: 餘因子 expansion by cofactors: 餘因子展開 upper triangular matrix: 上三角矩陣 lower triangular matrix: 下三角矩陣 diagonal matrix: 對角矩陣
3.20
3.2 Evaluation of a Determinant Using Elementary Row Operations
Theorem 3.3: Elementary row operations and determinants
,(a) ( ) det( ) det( )i jB I A B A
Let A and B be square matrices
( )(b) ( ) det( ) det( )kiB M A B k A
( ),(c) ( ) det( ) det( )k
i jB A A B A
Notes: The above three properties remains valid if elementary column operations are performed to derive column-equivalent matrices (This result will be used in Ex 5 on Slide 3.25)
The computational effort to calculate the determinant of a square matrix with a large number of n is unacceptable. In this section, I will show how to reduce the computational effort by using elementary operations
3.21
1 2 3
0 1 4 det( ) 2
1 2 1
A A
Ex:
1 1
4 8 12
0 1 4 det( ) 8
1 2 1
A A
2 2
0 1 4
1 2 3 det( ) 2
1 2 1
A A
3 3
1 2 3
2 3 2 det( ) 2
1 2 1
A A
(4)1 1
1
( )
det( ) 4det( ) (4)( 2) 8
A M A
A A
2 1, 2
2
( )
det( ) det( ) ( 2) 2
A I A
A A
( 2)3 1,2
3
( )
det( ) det( ) 2
A A A
A A
3.22
Row reduction method to evaluate the determinant 1. A row-echelon form of a square matrix is either an upper
triangular matrix or a matrix with zero rows 2. It is easy to calculate the determinant of an upper triangular
matrix (by Theorem 3.2) or a matrix with zero rows (det = 0)
Ex 2: Evaluation a determinant using elementary row operations
310221
1032A ?)det( A
Sol:
1,2
2 3 10 1 2 2
det( ) 1 2 2 2 3 10
0 1 3 0 1 3
IA
A 1A 1
1
det( ) det( )
det( ) det( )
A A
A A
Notes:
3.23
( 1)2,3
1 2 2
7 0 1 2 7( 1) 7
0 0 1
A
1( )( 2) 71,2 2
1 2 2 1 2 21
0 7 14 ( 1) ( ) 0 1 2(1/ 7)
0 1 3 0 1 3
A M
Notes:
2 1
3 2 2 3
4 3
det( ) det( )
1 1det( ) det( ) det( ) det( )
7 (1/ 7)
det( ) det( )
A A
A A A A
A A
2A 3A
4A
3.24
Cofactor Expansion Row Reduction
Order n Additions Multiplications Additions Multiplications
3 5 9 5 10
5 119 205 30 45
10 3,628,799 6,235,300 285 339
Comparison between the number of required operations for the two kinds of methods to calculate the determinant
※ When evaluating a determinant by hand, you can sometimes save steps by integrating this two kinds of methods (see Examples 5 and 6 in the next three slides)
3.25
Ex 5: Evaluating a determinant using column reduction and cofactor expansion
603
142
253
A
Sol:
( 2)1,3
3 1
3 5 2 3 5 4
det( ) 2 4 1 2 4 3
3 0 6 3 0 0
5 4( 3)( 1) ( 3)(1)( 1) 3
4 3
AC
A
※ is the counterpart column operation to the row operation ( ),k
i jAC ( ),k
i jA
3.26
Ex 6: Evaluating a determinant using both row and column reductions and cofactor expansion
Sol:
0231134213321011231223102
A
(1) ( 1)2,4 2,5
2 2
2 0 1 3 2 2 0 1 3 2
2 1 3 2 1 2 1 3 2 1
det( ) 1 0 1 2 3 1 0 1 2 3
3 1 2 4 3 1 0 5 6 4
1 1 3 2 0 3 0 0 0 1
2 1 3 2
1 1 2 3(1)( 1)
1 5 6 4
3 0 0 1
A A
A
3.27
( 3) (1)4,1 2,1
4 4
1 3
8 1 3 28 1 3 0 0 5
8 1 2 3(1)( 1) 8 1 2 = 8 1 2
13 5 6 413 5 6 13 5 6
0 0 0 1
8 1 5( 1)
13 5
(5)( 27)
135
AC A
3.28
Theorem 3.4: Conditions that yield a zero determinant
(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)
If A is a square matrix and any of the following conditions is
true, then det(A) = 0
Notes: For conditions (b) or (c), you can use elementary row or column operations to create an entire row or column of zeros
※ Thus, we can conclude that a square matrix has a determinant of zero if and only if it is row- (or column-) equivalent to a matrix that has at least one row (or column) consisting entirely of zeros
3.29
0
654
000
321
0
063
052
041
0
654
222
111
0
261
251
241
0
642
654
321
0
6123
5102
481
Ex:
3.30
3.3 Properties of Determinants
Notes:
)det()det()det( BABA
Theorem 3.5: Determinant of a matrix product
(1) det(EA) = det(E) det(A) ※ For elementary matrices shown in Theorem 3.3,
(3)
(4)
333231
232221
131211
333231
232221
131211
aaa
bbb
aaa
aaa
aaa
aaa
333231
232322222222
131211
aaa
bababa
aaa
det(AB) = det(A) det(B)
(There is an example to verify this property on Slide 3.33) (Note that this property is also valid for all rows or columns other than the second row)
,( )
( ),
if ( ), det( ) det( ) 1
if ( ), det( ) det( )
if ( ), det( ) 1det( ) 1
i jk
ik
i j
E I I E I
E M I E k I k
E A I E I
(2) 1 2 1 2det( ) det( )det( ) det( )n nA A A A A A
3.31
Ex 1: The determinant of a matrix product
101
230
221
A
213
210
102
B
7
101
230
221
||
A 11
213
210
102
|| B
Sol:
Find |A|, |B|, and |AB|
3.32
115
1016
148
213
210
102
101
230
221
AB
8 4 1
| | 6 1 10 77
5 1 1
AB
|AB| = |A| |B|
Check:
3.33
Ex:
?1 2 2 1 2 2 1 2 2
0 3 2 1 1 2 1 2 0
1 0 1 1 0 1 1 0 1
A B C
2 1 2 2 2 3
2 1 2 2 2 3
2 1 2 2 2 3
2 2 1 2 1 2| | 0( 1) 3( 1) 2( 1)
0 1 1 1 1 0
2 2 1 2 1 2| | 1( 1) 1( 1) 2( 1)
0 1 1 1 1 0
2 2 1 2 1 2| | 1( 1) 2( 1) 0( 1)
0 1 1 1 1 0
A
B
C
Pf:
3.34
Ex 2:
10 20 40 1 2 4
30 0 50 , if 3 0 5 5, find | |
20 30 10 2 3 1
A A
Sol:
132
503
421
10A 5000)5)(1000(
132
503
421
103
A
Theorem 3.6: Determinant of a scalar multiple of a matrix
If A is an n × n matrix and c is a scalar, then
det(cA) = cn det(A)(can be proven by repeatedly use the fact
that )
( )if ( ) kiB M A B k A
3.35
Theorem 3.7: (Determinant of an invertible matrix)
A square matrix A is invertible (nonsingular) if and
only if det(A) 0Pf:
If A is invertible, then AA–1 = I. By Theorem 3.5, we can have |A||A–1|=|I|. Since |I|=1, neither |A| nor |A–1| is zero
Suppose |A| is nonzero. It is aimed to prove A is invertible.
By the Gauss-Jordan elimination, we can always find a matrix B, in reduced row-echelon form, that is row-equivalent to A
1. Either B has at least one row with entire zeros, then |B|=0 and thus |A|=0 since |Ek|…|E2||E1||A|=|B|. →←
2. Or B=I, then A is row-equivalent to I, and by Theorem 2.15 (Slide 2.59), it can be concluded that A is invertible
( )
( )
3.36
Ex 3: Classifying square matrices as singular or nonsingular
0A
123
123
120
A
123
123
120
B
A has no inverse (it is singular)
012B B has inverse (it is nonsingular)
Sol:
3.37
Ex 4:?1 A
012
210
301
A?TA(a) (b)
4
012
210
301
|| A 4111
AA
4 AAT
Sol:
Theorem 3.8: Determinant of an inverse matrix
1 1If is invertible then det( )
det( )A A
A ,
Theorem 3.9: Determinant of a transpose
If is a square matrix, then det( ) det( )TA A A
1 1(Since , then 1)AA I A A
(Based on the mathematical induction (數學歸納法 ), compare the cofactor expansion along the row of A and the cofactor expansion along the column of AT)
3.38
The similarity between the noninvertible matrix and the real number 0
Matrix A Real number c
Invertible
Noninvertible
1 1
det( ) 0
1 exists and det( )
det( )
A
A AA
1
1
det( ) 0
does not exist
1 1det( )
det( ) 0
A
A
AA
1 1
0
1 exists and =
c
c cc
1
1
0
does not exist
1 1= =
0
c
c
cc
3.39
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) A is row-equivalent to In
(5) A can be written as the product of elementary matrices
(6) det(A) 0
Equivalent conditions for a nonsingular matrix:
※ The statements (1)-(5) are collected in Theorem 2.15, and the statement (6) is from Theorem 3.7
(Thm. 2.11)
(Thm. 2.11)
(Thm. 2.14)
(Thm. 2.14)
(Thm. 3.7)
3.40
Ex 5: Which of the following system has a unique solution?
(a)
423
423
12
321
321
32
xxx
xxx
xx
(b)
423
423
12
321
321
32
xxx
xxx
xx
3.41
Sol:
(the coefficient matrix is the matrix in Ex 3)A Ax b(a)
0 (from Ex 3)A This system does not have a unique solution
(b) (the coefficient matrix is the matrix in Ex 3)B Bx b
12 0 (from Ex 3)B
This system has a unique solution
3.42
3.4 Applications of Determinants
Matrix of cofactors (餘因子矩陣 ) of A:
nnnn
n
n
ij
CCC
CCC
CCC
C
21
22221
11211
ijji
ij MC )1(
11 21 1
12 22 2
1 2
adj( )
n
T nij
n n nn
C C C
C C CA C
C C C
Adjoint matrix (伴隨矩陣 ) of A:
(The definition of cofactor Cij and minor Mij of aij can be reviewed on Slide 3.6)
11 12 111 1 1
12 2 2
1 2
1
1 2
det( ) 0 0
0 det( ) 0
0 0 det( )
nj n
j n
i i in
n jn nn
n n nn
a a aC C C A
C C C Aa a a
C C C Aa a a
3.43
Theorem 3.10: The inverse of a matrix expressed by its adjoint matrix
If A is an n × n invertible matrix, then 1 1adj( )
det( )A A
A
Consider the product A[adj(A)]
1 1 2 2
det( ) if
0 if i j i j in jn
A i ja C a C a C
i j
The entry at the position (i, j) of A[adj(A)]
Pf:
3.44
Consider a matrix B similar to matrix A except that the j-th row is replaced by the i-th row of matrix A
11 12 1
1 2
1 2
1 2
det( ) 0
n
i i in
i i in
n n nn
a a a
a a a
B
a a a
a a a
1 1 2 2det( ) 0i j i j in jnB a C a C a C
Perform the cofactor expansion along the j-th row of matrix B
(Note that Cj1, Cj2,…, and Cjn are still the cofactors of aij)
1[adj( )] det( ) adj( )
det( )A A A I A A I
A
A-1
※ Since there are two identical rows in B, according to Theorem 3.4, det(B) should be zero
3.45
det( ) ,A ad bc
dc
baA
11 21
12 22
adj( )C C d b
AC C c a
1 1
adj( )det
A AA
Ex: For any 2×2 matrix, its inverse can be calculated as follows
ac
bd
bcad1
3.46
Ex 2:
201
120
231
A(a) Find the adjoint matrix of A
(b) Use the adjoint matrix of A to find A–1
11
2 14,
0 2C
Sol:
12
0 11,
1 2C
13
0 22,
1 0C
ijji
ij MC )1(
21
3 26,
0 2C
22
1 20,
1 2C
23
1 33,
1 0C
31
3 27,
2 1C
32
1 21,
0 1C
33
1 32.
0 2C
3.47
cofactor matrix of A
217
306
214
ijC
adjoint matrix of A
4 6 7
adj( ) 1 0 1
2 3 2
T
ijA C
232
101
764
31
inverse matrix of A
1 1
adj( )det
A AA
(det 3)A
32
32
31
31
37
34
1
0
2
Check: IAA 1
※ The computational effort of this method to derive the inverse of a matrix is higher than that of the G.-J. E. (especially to compute the cofactor matrix for a higher-order square matrix)
※ However, for computers, it is easier to implement this method than the G.-J. E. since it is not necessary to judge which row operation should be used and the only thing needed to do is to calculate determinants of matrices
3.48
Theorem 3.11: Cramer’s Rule
11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
n n
n n
n n nn n n
a x a x a x b
a x a x a x b
a x a x a x b
A x b
(1) (2) ( )where ,nij n n
A a A A A
1
2 ,
n
x
x
x
x
1
2
n
b
b
b
b
11 12 1
21 22 2
1 2
Suppose this system has a unique solution, i.e.,
det( ) 0
n
n
n n nn
a a a
a a aA
a a a
A(i) represents the i-th column vector in A
3.49
(1) (2) ( 1) ( 1) ( )By defining j j njA A A A A A b
11 1( 1) 1 1( 1) 1
21 2( 1) 2 2( 1) 2
1 ( 1) ( 1)
j j n
j j n
n n j n n j nn
a a b a a
a a b a a
a a b a a
1 1 2 2(i.e., det( ) )j j j n njA b C b C b C
det( ), 1, 2, ,
det( )j
j
Ax j n
A
3.50
Pf:
(det( ) 0)A Ax = b
bx 1 A1
adj( )det( )
AA
b
11 21 1 1
12 22 2 2
1 2
1
det( )
n
n
n n nn n
C C C b
C C C b
A
C C C b
1 11 2 21 1
1 12 2 22 2
1 1 2 2
1
det( )
n n
n n
n n n nn
b C b C b C
b C b C b C
A
b C b C b C
(according to Thm. 3.10)
3.51
1 1 11 2 21 1
2 1 12 2 22 2
1 1 2 2
1
2
1
det( )
det( ) / det( )
det( ) / det( )
det( ) / det( )
n n
n n
n n n n nn
n
x b C b C b C
x b C b C b C
A
x b C b C b C
A A
A A
A A
det( ), 1, 2, ,
det( )j
j
Ax j n
A
1 1 2 2
(On Slide 3.49, it is already derived that
det( ) )j j j n njA b C b C b C
3.52
Ex 6: Use Cramer’s rule to solve the system of linear equation
2443
02
132
zyx
zx
zyx
Sol:
8
442
100
321
)det( 1
A10
443
102
321
)det(
A
,15
423
102
311
)det( 2
A 16
243
002
121
)det( 3
A
54
)det()det( 1
AA
x23
)det()det( 2
AA
y58
)det()det( 3
AA
z
3.53
Keywords in Section 3.4:
matrix of cofactors: 餘因子矩陣 adjoint matrix: 伴隨矩陣 Cramer’s rule: Cramer 法則
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