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Determinants, Eigenvaluesand Eigenvectors

Lecture:

1 The Determinant of a Matrix

2 Evaluation of a Determinant using Elementary

Operations

By

Dr. Mai Duc Thanh

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3.1 The Determinant of a Matrix

3.2 Evaluation of a Determinant using

Elementary Operations

3.3 Properties of Determinants 3.4 Cramers Rule

3.5 Introduction to Eigenvalues

Determinants, Eigenvalues and Eigenvectors

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The determinant of a 2 2 matrix:

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3.1 The Determinant of a Matrix

Note:

2221

1211

aa

aaA

12212211||)det( aaaaAA

2221

1211

aaaa

2221

1211

aaaa

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Ex. 1: (The determinant of a matrix of

order 2))3(1)2(2

21

32

24

12

42

30

Note: The determinant of a matrix can be positive, zero, or negative.

34 7

)1(4)2(2 44 0

)3(2)4(0 60 6

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Cofactor of :

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

Minor of the entry :

ij

ji

ij MC )1(

ija

ija

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Ex:

333231

232221

131211

aaa

aaaaaa

A

3332

131221

aa

aaM

2121

12

21 )1( MMC

3331

131122

aaaaM

2222

22

22 )1( MMC

Notes: Sign pattern for cofactors

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Definition: (Expansion by cofactors)

n

j

ininiiiiijij CaCaCaCaAAa

1

2211||)det()(

(Cofactor expansion along the i-th row, i=1, 2,, n)

n

i

njnjjjjjijij CaCaCaCaAAb1

2211||)det()(

(Cofactor expansion along the j-th row, j=1, 2,, n )

LetA isasquare matrix of ordern.

Then the determinant ofA is given by

or

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Ex: The determinant of a matrix of order 3

333323231313

323222221212

313121211111

333332323131

232322222121

131312121111)det(

CaCaCa

CaCaCa

CaCaCa

CaCaCa

CaCaCaCaCaCaA

333231

232221

131211

aaa

aaaaaa

A

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Ex 5: (The determinant of a matrix of order 3)

?)det( A

1 1

11

1 2( 1) 1

0 1C

0 2 13 1 2

4 0 1

A

Sol:

5)5)(1(14

23)1( 2112

C

4

04

13)1( 3113

C

14

)4)(1()5)(2()1)(0(

)det( 131312121111

CaCaCaA

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Ex 4: (The determinant of a matrix of

order 4)

?)det( A

2043

3020

2011

0321

A

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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

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The determinant of a matrix of order 3:

333231

232221

131211

aaa

aaa

aaa

A

3231333231

2221232221

1211131211

aaaaa

aaaaa

aaaaa

122133112332

132231322113312312332211

||)det(

aaaaaa

aaaaaaaaaaaaAA

Add these three products.

Subtract these three products.

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Ex 5:

144

213

120

A

44

13

20

4

0

260)4(12160||)det( AA

16 12

0 6

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4 - 14

Upper triangular matrix:

Lower triangular matrix:

Diagonal matrix:

All the entries below the main diagonal are zeros.

All the entries below the main diagonal are zeros.

All the entries above and below the main diagonal are zeros.

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33

2322

131211

00

0

a

aa

aaa

333231

2221

11

0

00

aaa

aa

a

33

22

11

00

00

00

a

a

a

Ex:

upper triangular lower triangular diagonal

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Determinant of a Triangular

Matrix

Thm 3.2:

IfA is an nxn 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(

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Ex 6: Find the determinants of the following triangular matrices.

(a)

3351

0165

0024

0002

A (b)

2000004000

00200

00030

00001

B

|A| = (2)(2)(1)(3) =12

|B| = (1)(3)(2)(4)(2) = 48

(a)

(b)

Sol:

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Keywords in Section 3.1:

determinant

minor

cofactor

expansion by cofactors

upper triangular matrix

lower triangular matrix

diagonal matrix

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Row elementary matrix:

An nn matrix is called an elementary matrix if it can be obtained

from the identity matrixI by a single elementary operation.

Three row elementary matrices:

)()1( IrR ijij

)0()()2()()( kIrR ki

k

i

)()3( )()( IrR kijk

ij

Interchange two rows.

Multiply a row by a nonzero constant.

Add a multiple of a row to anotherrow.

Note:

Only do a single elementary row operation.

3.2 Evaluation of a determinant

using elementary operations

3 2 Evaluation of a determinant using elementary

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3.2 Evaluation of a determinant using elementaryoperations

Thm 3.3: (Elementary row operations anddeterminants)

)()( ArBa ij )det()det( AB

LetA andB be square matrices.

)()()(

ArBbk

i )det()det( AkB

)()()(

ArBck

ij )det()det( AB

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121

410321

A

Ex:

121

410

1284

1A

121

321

410

2A

121

232

321

3A

2)det( A

8)2)(4()det(4)det()( 1)4(

11

AAArA

2)2()det()det()( 2122 AAArA

2)det()det()( 3)2(123 AAArA

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?)det( A

Note:

A row-echelon form of a square matrix is always upper triangular.

Ex 2: (Evaluation a determinant using elementary row operations)

310221

1032A

Sol:

3101032

221

310221

1032)det( 12

rA

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Notes:

7)1)(1)(1)(7(

100

210221

7)1(

23

r

))(det(1)det( )( Ark

A ki

310210

221

)1

)(1(310

1470

221

71

)71(

2)2(

12

rr

))(det()det( ArA ij

))(det()det()(

ArAk

ij

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Notes:

AEEA

ijRE)1( 1 ijRE

AEARAArEA ijij

)()2( kiRE kREk

i

AEARAkArEA kik

i )(

)()3( kijRE 1k

ijRE

AEARAArEA kijk

ij )()( 1

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Thm 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).

IfA is a square matrix and any of the following conditions is true,

then det (A) = 0.

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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:

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Ex 5: (Evaluating a

determinant)

3)

5

3)(5(

63)1)(5(

603

0253

603142

253)det(

53

52

21

53

52

)54(

12

CA

603

142253

A

Sol:

3)1)(3(3445

)1)(3(

003342453

603142

253)det(

13

)2(13

CA

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Ex 6: (Evaluating a

determinant)

Sol:

02311342133210112312

23102

A

(1)24( 1)25

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

r

r

A

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( 3)41

(1)21

4 4

8 1 3 2

8 1 3 0 0 58 1 2 3(1)( 1) 8 1 2 8 1 2

13 5 6 413 5 6 13 5 6

0 0 0 1

C

r

135

)27)(5(

513

181)5( 31

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Evaluate the determinants

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Exercises

2 1 3 2 4 4 1 3

3 0 4 5 2 3 1 1

det( ) , det( )4 2 5 1 2 1 3 5

1 2 3 2 1 3 0 4

A B

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Exercises

12

4 3

2 1 3 2 1 2 3 2

3 0 4 5 0 3 4 5det( )

4 2 5 1 2 4 5 1

1 2 3 2 2 1 3 2

1 2 3 2 1 2 3 2

0 3 4 5 0 3 4 5

2 4 5 1 0 0 1 5

0 3 2 1 0 3 2 1

3 4 5 0 6 60 1 5 0 1 5 3( 30 6)

3 2 1 3 2 1

C

R R

A

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Textbook: R. Hill, Elementary LinearAlgebra with Applications, 3rd Edition

Page 122: 3, 4, 9, 11.14, 18, 19, 20, 21, 22

Deadline: 16th March, 2010

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Homework: Determinants