chapter 3 determinants
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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. ※ The determinant is NOT a matrix operation - PowerPoint PPT PresentationTRANSCRIPT
Chapter 3Determinants
3.1 The Determinant of a Matrix 3.2 Evaluation of a Determinant using Elementary
Row Operations3.3 Properties of Determinants 3.4 Application of Determinants: Cramer’s Rule
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 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
3.3
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
aaaa
A
12212211||)det( aaaaAA
2221
1211
aaaa
2221
1211
aaaa
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 ba x a x b
b a b a b a b ax xa a a a a a a a
3.5
Ex. 1: The determinant of a matrix of order 2
734)3(1)2(22132
044)1(4)2(22412
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
aaaaaaaa
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
aaaaaaaaa
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
aaaaaaaaa
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 Ca C a C a Ca C a C a Ca C a C a Ca C a C a
2 32
13 13 23 23 33 33
(second column expansion) (third column expansion)
Ca C a C a C
3.10
Ex 3: The determinant of a square matrix of order 3
?)det( A0 2 13 1 24 0 1
A
11021
)1( 1111
C
Sol:
5)5)(1(1423
)1( 2112 C
40413
)1( 3113
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 aA a a a
a a a
3231333231
2221232221
1211131211
aaaaaaaaaaaaaaa
122133112332
132231322113312312332211
||)det(
aaaaaaaaaaaaaaaaaaAA
Add these three products
Subtract these three products
3.12
Ex: Recalculate the determinant of the square matrix A in Ex 3
0 2 13 1 24 0 1
A
0 23 14 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
2043302020110321
A ?)det( A
3.14
Sol:))(0())(0())(0())(3()det( 43332313 CCCCA
243320211
)1(3 31
133C
39)13)(3(
)7)(1)(3()4)(1)(2(034311
)1)(3(23
21)1)(2(
2421
)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
11000
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(
※ Based on Theorem 3.2, it is straightforward to obtain that※ 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
det( ) 1I
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)
3351016500240002
A (b)
2000004000002000003000001
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 30 1 4 det( ) 21 2 1
A A
Ex:
1 1
4 8 120 1 4 det( ) 81 2 1
A A
2 2
0 1 41 2 3 det( ) 21 2 1
A A
3 3
1 2 32 3 2 det( ) 2
1 2 1A A
(4)1 1
1
( )det( ) 4det( ) (4)( 2) 8A M A
A A
2 1, 2
2
( )det( ) det( ) ( 2) 2A 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 2det( ) 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 27 0 1 2 7( 1) 7
0 0 1
A
1( )( 2) 71,2 2
1 2 2 1 2 210 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
603142
253A
Sol:
( 2)1,3
3 1
3 5 2 3 5 4det( ) 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 22 1 3 2 1 2 1 3 2 1
det( ) 1 0 1 2 3 1 0 1 2 33 1 2 4 3 1 0 5 6 41 1 3 2 0 3 0 0 0 1
2 1 3 21 1 2 3
(1)( 1)1 5 6 43 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 also use elementary row or column operations to create an entire row or column of zeros and obtain the results by Theorem 3.3 ※ 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
(For (b) and (c), based on the mathematical induction (數學歸納法 ), perform the cofactor expansion along any row or column other than these two rows or columns)
(Perform the cofactor expansion along the zero row or column)
3.29
0654000321
0063052041
0654222111
0261251241
0642
654321
061235102481
Ex:
3.30
3.3 Properties of Determinants
Notes:
)det()det()det( BABA
Theorem 3.5: Determinant of a matrix product
(1)
(3)
(4)11 12 13 11 12 13
21 22 23 21 22 23
31 32 33 31 32 33
a a a a a aa a a b b ba a a a a a
11 12 13
21 21 22 22 23 23
31 32 33
a a aa b a b a b
a a a
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)
, , , ,( ) ( ) ( ) ( )
( ) ( ) ( ) ( ), , , ,
det( ) det( )det( ) and det( ) det( ) det( ) 1 det( ) det( )det( ) and det( ) det( ) det( )det( ) det( )det( ) and det( ) det( ) det( ) 1
i j i j i j i jk k k k
i i i ik k k k
i j i j i j i j
E A E A E A A EE A E A E A k A E kE A E A E A A E
(2)
1 2 1 2det( ) det( )det( ) det( )n nA A A A A A
(Verified by Ex 1 on the next slide)
3.31
Ex 1: The determinant of a matrix product
101230221
A
213210
102B
7101230221
||
A 11213210
102||
B
Sol:
Find |A|, |B|, and |AB|
3.32
1151016148
213210
102
101230221
AB
8 4 1 | | 6 1 10 77
5 1 1AB
|AB| = |A| |B| Check:
3.33
Ex:
?1 2 2 1 2 2 1 2 20 3 2 1 1 2 1 2 01 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 2 430 0 50 , if 3 0 5 5, find | |20 30 10 2 3 1
A A
Sol:
132503421
10A 5000)5)(1000(132503421
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) 0
Pf:
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
123123120
A
123123120
B
A has no inverse (it is singular)
012B B has inverse (it is nonsingular)
Sol:
3.37
Ex 4:?1 A
012210301
A?TA(a) (b)
4012210301
|| A 4111
AA
4 AAT
Sol:
Theorem 3.8: Determinant of an inverse matrix1 1If is invertible then det( )
det( )A A
A ,
Theorem 3.9: Determinant of a transposeIf 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( ) 01 exists and det( )
det( )
A
A AA
1
1
det( ) 0
does not exist
1 1det( )det( ) 0
A
A
AA
1 1
01 exists and =
c
c cc
1
1
0
does not exist1 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)
42342312
321
321
32
xxxxxxxx
(b)
42342312
321
321
32
xxxxxxxx
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 b12 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
CCCCCC
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 CC C C
A C
C C C
Adjoint matrix (伴隨矩陣 ) of A:
(The definitions of cofactor Cij and minor Mij of aij can be reviewed on Slide 3.6)
11 12 111 1 1
12 2 21 2
11 2
det( ) 0 00 det( ) 0
0 0 det( )
nj n
j ni i in
n jn nnn n nn
a a aC C C AC C C A
a 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 1 adj( )det( )
A AA
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 aB
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 for the entries of the j-th row)
1[adj( )] det( ) adj( )det( )
A A A I A A IA
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
dcba
A
11 21
12 22
adj( )C C d b
AC C c a
1 1 adj( )
detA A
A
Ex: For any 2×2 matrix, its inverse can be calculated as follows
acbd
bcad1
3.46
Ex 2:
201120231
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
217306214
ijC
adjoint matrix of A
4 6 7adj( ) 1 0 1
2 3 2
T
ijA C
232101764
31
inverse matrix of A
1 1 adj( )
detA A
A (det 3)A
32
32
31
31
37
34
102
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 ba 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
xx
x
x
1
2
n
bb
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 aa a a
A
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 aa 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( )A
A b
11 21 1 1
12 22 2 2
1 2
1det( )
n
n
n n nn n
C C C bC C C b
AC C C b
1 11 2 21 1
1 12 2 22 2
1 1 2 2
1det( )
n n
n n
n n n nn
b C b C b Cb C b C b C
Ab 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 Cx b C b C b C
Ax b C b C b C
A AA 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
244302132
zyxzxzyx
Sol:
8442100321
)det( 1
A10
443102321
)det(
A
,15423102311
)det( 2
A 16243002121
)det( 3
A
54
)det()det( 1
AAx
23
)det()det( 2
AAy
58
)det()det( 3
AAz
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