Chap. 3 Determinants

Chap. 3Chap. 3DeterminantsDeterminants

3.1 The Determinants of a Matrix3.2 Evaluation of a Determinant Using Elementary Operations3.3 Properties of Determinants3.4 Introduction to Eigenvalues3.5 Applications of Determinants

Ming-Feng Yeh Chapter 3 3-2

Every square matrix can be associated with a real number called its determinant.

Definition: The determinant of the matrix

is given by

Example 1:

3.1 3.1 The Determinant of a MatrixThe Determinant of a Matrix

2221

1211

aaaa

A

122122112221

1211)det( aaaaaaaa

AA

+

?4230

?2412

?2132

2(2) 1(3) = 7

2(2) 1(4) = 0

0(4) 2(3) = 6

?]2[ AA 2


Minors and Cofactors of a MatrixMinors and Cofactors of a Matrix If A is a square matrix, then the minor (子行列式 ) Mij of

the element aij is the determinant of the matrix obtained by deleting the ith row and jth column of A.The cofactor (餘因子 ) Cij is given by Cij = (1)i+jMij.

Sign pattern for cofactors:

Section 3-1

333231

232221

131211

aaaaaaaaa

A 3332

131221 aa

aaM 2121

1221 )1( MMC

3331

131122 aa

aaM 2222

2222 )1( MMC

4433


Theorem 3.1Theorem 3.1Expansion by Cofactors Let A be a square matrix of order n. Then the determinant

of A is given by

For any 33 matrix:

ininiiii

n

jijij CaCaCaCaAA

2211

1)det(

njnjjjjj

n

iijij CaCaCaCaAA

2211

1)det(

ith row expansion

jth column expansion

333231

232221

131211

aaaaaaaaa

A

122133112332132231322113312312332211 aaaaaaaaaaaaaaaaaaA

333231

232221

131211

aaaaaaaaa

3231

2221

1211

aaaaaa

+

+ +

Section 3-1


Examples 2 & 3Examples 2 & 3 Find all the minors and cofactors of A, and then find the

determinant of A.Sol:

104213120

A

,11021

11

M ,51423

12 M 40413

13

M

.6,3,5,8,4,2

333231

232221

MMMMMM

111 C 512 C 413 C

.6,3,5,8,4,2

333231

232221

CCCCCC

14)5(4)2(3)1(014)8(2)4)(1()2(3

14)4(1)5(2)1(0

313121211111

232322222121

131312121111

CaCaCaCaCaCa

CaCaCaA

Section 3-1


Example 5Example 5 Find the determinant of Sol:

Section 3-1

144213120

A

144213120

441320

+(0) +(16)

(4)

+(12)

(6)(0)

260)4()12(160 A

144213120


Example 4Example 4 Find the determinant ofSol: Expansion by which row

or which column? the 3rd column: three of the entires are zeros

2043302020110321

A

243320211

243320211

)1( 3113

C

13)7)(3()4)(2(04311

)1)(3(2321

)1)(2(2421

)1)(0( 322212

13121294)2)(1(0)1)(3(4)2)(2(3)3)(3(1)2)(4(0)2)(2)(1(

39)13(31313 CaA

Section 3-1


Triangular MatricesTriangular MatricesUpper triangular Matrix Lower triangular Matrix

Theorem 3.2: If A is a triangular matrix of order n, then its determinant is the product of the entires on the main diagonal. That is,

nn

n

n

n

a

aaaaaaaaa

000

000

333

22322

1131211

nnnnn aaaa

aaaaa

a

321

333231

2221

11

000000

nnaaaaAA 332211)det(

Section 3-1


ExampleExample

?300210132

3021

)1(2 11 6)]2(0)3)(1[(2

?

2000004000002000003000001

?

3351016500240002

12)3)(1)(2(2

48)2)(4)(2)(3)(1(

Section 3-1


3.2 Evaluation of a Determinant3.2 Evaluation of a Determinant Using Elementary Operations Using Elementary Operations Which of the following two determinants is easier to evaluate?

2963394223641321

A

1000130029201321

B

6)18()10(3)39(2)60(1

963942364

)1(1263342264

3

293392234

)1)(2(296394236

1

6)1)(3)(2)(1(

By elementary row operations


Theorem 3.3Theorem 3.3Elementary Row Operations and Determinants Let A and B be square matrices.1. If B is obtained from A by interchanging two rows of A,

then det(B) = det(A).2. If B is obtained from A by adding a multiple of a row of A

to another row of A, then det(B) = det(A).3. If B is obtained from A by multiplying a row of A by a

nonzero constant c, then det(B) = cdet(A).

23412

A 21234

.1 B 21012

.2 B

(2)

36

3436

.3 B

Take a common factor out of a row

Section 3-2


Example 2Example 2 Find the determinant ofSol:

3101032

221

310221

1032A

310221

1032

3101470

221

(2)

Factor 7 out of the 2nd row

310210221

7

(1) 100

210221

7

7)1)(1)(1(7

Section 3-2


Determinants andDeterminants andElementary Column Operations Elementary Column Operations Although Theorem 3.3 was stated in terms of elementary

row operations, the theorem remains valid if the word “row” is replaced by the word “column.”

Operations performed on the column of a matrix are called elementary column operations.

Two matrices are called column-equivalent if one can be obtained from the other by elementary column operations.

Section 3-2



3105463221

A

3105463221

(2)

305403201

Expansion by the second column

0)0()0()0( 322212 CCC

Section 3-2


Theorem 3.4Theorem 3.4Conditions That Yield a Zero Determinant If A is a square matrix and any one of the following

conditions is true, then det(A) = 0.1. An entire row (or an entire column) consists of zeros.2. Two rows (or columns) are equal.3. One row (or column) is a multiple of another row (or

column).

0421210421

0602612321

0253542000

(1)

(3)

Section 3-2


Examples 4 & 5Examples 4 & 5

?4180012141

?603142253

4180290141

(2)

0

(2)

003342453

3445

)1(3 13

3)1)(1(3

Section 3-2



0231134213321011231223102

A (1)

1000346501321011231223102

A

1003465132112312

)1)(1( 22

(3)

10004651332182318

6513218318

)1)(1( 44

6513218500

51318

)1(5 22

135)1340(5

Section 3-2


3.3 Properties of Determinants3.3 Properties of Determinants Example 1: Find for the matrices

Sol:

ABBA and,,

213210102

and101230221

BA

11213210102

and7101230221

BA

1151016

148

213210102

101230221

AB

77 BA

77 AB


Theorems 3.5 & 3.6 Theorems 3.5 & 3.6 Theorem 3.5: Determinant of a Matrix Product If A and B are square matrices of order n, then

det(AB) = det(A) det(B)Remark:

Theorem 3.6: Determinant of a Scalar Multiple of a Matrix If A is a nn matrix and c is a scalar, then the determinant

of cA is given by det(cA) = cn det(A).Remark: [Thm. 3.3] If B is obtained from A by multiplying a

row of A by a nonzero constant c, then det(B) = cdet(A).

kk AAAAAAAA 321321 BABA

Section 3-3


Example 2Example 2 Find the determinant of the matrixSol:

10302050030402010

A

132503421

10A 5132503421

5000)5(1000132503421

103

A

21226

AA

31073

BB18

0299

BABA

BA

Section 3-3


Theorems 3.7 & 3.8Theorems 3.7 & 3.8Theorem 3.7: Determinant of an Invertible Matrix A square matrix A is invertible (nonsingular) if and only if

det(A) 0.Theorem 3.8: Determinant of an Inverse Matrix If A is invertible, then det(A1) = 1 / det(A).

Hint: A is invertible AA1 = I

11 IAA

Section 3-3


Example 3 & 4Example 3 & 4Example 3: Which of the matrices has an inverse?

Sol:

Example 4: Find for the matrix Sol:

123123120

123123120

BA

singular)(0A

It has no inverse.ar)(nonsingul012 B

It has an inverse.1A

012213301

A 4A4111

AA

Section 3-3


Equivalent Conditions for aEquivalent Conditions for a Nonsingular Matrix Nonsingular Matrix If A is an nn matrix, then the following statements are

equivalent.1. A is invertible.2. Ax = b has a unique solution for every n1 column vector b.3. Ax = O has only the trivial solution.4. A is row-equivalent to In.5. A can be written as the product of elementary matrices.

【 Also see in Theorem 2.15 】6. det(A) 0.

【 See Example 5 (p.148) for instance 】

Section 3-3


Determinant of a TransposeDeterminant of a TransposeTheorem 3.9: If A is a square matrix, then det(A)=det(AT).

Example 6: Show that for the following matrix. pf:

514002213

A

TAA

6)3)(2(5121

)1)(2( 12

A

502101423

TA 6)3)(2(5211

)1)(2( 21

TA

. Thus, TAA

Section 3-3


3.4 Introduction to Eigenvalues3.4 Introduction to Eigenvalues

See Chapter 7


3.5 Applications of 3.5 Applications of DeterminantsDeterminants The Adjoint of a Matrix

If A is a square matrix, then the matrix of cofactors of A

has the form

The transpose of this matrixis called the adjoint of A andis denoted by adj(A).

nnnn

n

n

CCC

CCCCCC

21

22221

11211

nnnn

n

n

CCC

CCCCCC

A

21

22212

12111

)(adj


Example 1Example 1 Find the adjoint ofSol:

The matrix of cofactors of A:

201120231

A

2031

1021

1223

0131

2121

2023

0120

2110

2012

217306214

232101764

)(adj A

201120231

012031

2012031231

Section 3-5


Theorem 3.10Theorem 3.10The Inverse of a Matrix Given by Its Adjoint If A is an nn invertible matrix, then

If A is 22 matrix

then the adjoint of A is .

Form Theorem 3.10 you have

)(adj)det(

11 AA

A

Section 3-5

,

dcba

A

acbd

A)(adj

acbd

bcadA

AA 1)(adj11


Example 2Example 2 Use the adjoint of to find .

Sol:

201120231

A 1A

232101764

)(adj A

3)2)(2)(1()1)(1)(3()2)(2)(1( A

32

32

31

31

37

34

1

102

232101764

31)(adj

A1 AA

?Check 1 IAA

Section 3-5


Theorem 3.11: Cramer’s RuleTheorem 3.11: Cramer’s Rule If a system of n linear equations in n variables has a

coefficient matrix with a nonzero determinant ,then the solution of the system is given by

where the ith column of Ai is the column of constants in the system of equations.

A

,)det()det(,,

)det()det(,

)det()det( 2

21

1 AAx

AAx

AAx n

n

3333232131

2323222121

1313212111

bxaxaxabxaxaxabxaxaxa

333231

232221

131211

33231

22221

112113

3

aaaaaaaaa

baabaabaa

AA

x

Section 3-5


Example 4Example 4 Use Cramer’s Rule to solve the system of linear equation

for x.

Sol:244302132

zyxzxzyx

10443102321

A solution) uniquean has system (the0

10442100321

1

AA

x54

108

10)1)(1)(4()2)(1)(2(

58,

23

zy

Section 3-5


Area of a TriangleArea of a Triangle The area of a triangle whose vertices

are (x1, y1), (x2, y2), and (x3, y3) is

given by

where the sign () is chosen to give a positive area.pf: Area =

111

21Area

33

22

11

yxyxyx

))(( 133121 xxyy ))(( 32232

1 xxyy ))(( 122121 xxyy

)( 23123113322121 yxyxyxyxyxyx

111

21

33

22

11

yxyxyx

Section 3-5


Example 5Example 5Fine the area of the triangle whose vertices are (1, 0), (2, 2),

and (4, 3).Sol:

Fine the area of the triangle whosevertices are (0, 1), (2, 2), and (4, 3).

23

134122101

21

23

Area

(1,0)

(2,2)

(4,3)

0Area0134122110

21

Three points in the xy-plane lie on the same line.

Section 3-5


Collinear Pts & Line EquationCollinear Pts & Line Equation Test for Collinear Points in the xy-Plane

Three points (x1, y1), (x2, y2), and (x3, y3) are collinear

if and only if

Two-Point Form of the Equation of a LineAn equation of the line passing through the distinct points

(x1, y1) and (x2, y2) is given by

0111

33

22

11

yxyxyx

0111

22

11 yxyxyx

The 3rd point: (x, y)

Section 3-5


Example 6Example 6 Find an equation of the line passing through the points

(2, 4) and (1, 3).Sol:

01311421

yx

1314

x1112

y 0

3142

1

0103 yx

An equation of the line is x 3y = 10.

Section 3-5


Volume of TetrahedronVolume of Tetrahedron The volume of the tetrahedron

whose vertices are (x1,y1, z1), (x2, y2, z2), (x3, y3, z3), and (x4, y4, z4), is given by

where the sign () is chosen to give a positive area.

Example 7: Find the volume of the tetrahedron whose vertices are (0,4,1), (4,0,0), (3,5,2), and (2,2,5).

Sol:

1111

61Volume

444

333

222

111

zyxzyxzyxzyx

12)72(

61

1522125310041140

61

12Volume

Section 3-5


Coplanar Pts & Plane EquationCoplanar Pts & Plane Equation Test for Coplanar Points

in SpaceFour points (x1,y1, z1), (x2, y2, z2), (x3, y3, z3), and (x4, y4, z4) are coplanar if and only if

Three-Point Form of the Equation of a PlaneAn equation of the plane passing through the distinct points (x1,y1, z1), (x2, y2, z2), and (x3, y3, z3) is given by

0

1111

444

333

222

111

zyxzyxzyxzyx

0

1111

333

222

111

zyxzyxzyxzyx

Section 3-5


Example 8Example 8 Find an equation of the plane passing through the points

(0,1,0), (1,3,2) and (2,0,1).Sol:

0

1102123110101

zyx

(1)

0

11121221100011

zyx

0112221

1

zyx3534 zyx

Section 3-5

Chap. 3 Determinants

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