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