Chapter 7: Matrices and Systems of Equations and Inequalities

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Copyright © 2007 Pearson Education, Inc. Slide 7-2

Chapter 7: Matrices and Systems of Equations and Inequalities

7.1 Systems of Equations

7.2 Solution of Linear Systems in Three Variables

7.3 Solution of Linear Systems by Row Transformations

7.4 Matrix Properties and Operations

7.5 Determinants and Cramer’s Rule

7.6 Solution of Linear Systems by Matrix Inverses

7.7 Systems of Inequalities and Linear Programming

7.8 Partial Fractions

Copyright © 2007 Pearson Education, Inc. Slide 7-3

• Subscript notation for the matrix A

The row 1, column 1 element is a11; the row 2, column 3 element is a23; and, in general, the row i, column j element is aij.

7.5 Determinants and Cramer’s Rule

mnmmm

n

n

n

aaaa

aaaaaaaaaaaa

A

321

3333231

2232221

1131211

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7.5 Determinants of 2 × 2 Matrices

• Associated with every square matrix is a real number called the determinant of A. In this text, we use det A.

The determinant of a 2 × 2 matrix A,

is defined as

2221

1211

aaaaA

.aaaaA 12212211 det

Copyright © 2007 Pearson Education, Inc. Slide 7-5

Example Find det A if

Analytic Solution Graphing Calculator Solution

7.5 Determinants of 2 × 2 Matrices

.8643

A

48

)4(6)8(3

8643det det

A

Copyright © 2007 Pearson Education, Inc. Slide 7-6

7.5 Determinant of a 3 × 3 Matrix

The determinant of a 3 × 3 matrix A,

is defined as

333231

232221

131211

aaaaaaaaa

A

).aaaaaaaa(aaaaaaaaaaA

122133112332132231

322113312312332211

det

Copyright © 2007 Pearson Education, Inc. Slide 7-7

• A method for calculating 3 × 3 determinants is found by re-arranging and factoring this formula.

Each of the quantities in parentheses represents the determinant of a 2 × 2 matrix that is part of the 3 × 3 matrix remaining when the row and column of the multiplier are eliminated.

7.5 Determinant of a 3 × 3 Matrix

)()()(

det

1322231231

13323312212332332211

333231

232221

131211

aaaaaaaaaaaaaaa

aaaaaaaaa

A

Copyright © 2007 Pearson Education, Inc. Slide 7-8

7.5 The Minor of an Element

• The determinant of each 3 × 3 matrix is called a minor of the associated element.

• The symbol Mij represents the minor when the ith row and jth column are eliminated.

Copyright © 2007 Pearson Education, Inc. Slide 7-9

• To find the determinant of a 3 × 3 or larger square matrix:1. Choose any row or column,2. Multiply the minor of each element in that row or

column by a +1 or –1, depending on whether the sum of i + j is even or odd,

3. Then, multiply each cofactor by its corresponding element in the matrix and find the sum of these products. This sum is the determinant of the matrix.

7.5 The Cofactor of an Element

Let Mij be the minor for element aij in an n × n matrix. The cofactor of aij, written Aij, is

.MA ijji

ij 1

Copyright © 2007 Pearson Education, Inc. Slide 7-10

Example Evaluate det , expanding

by the second column.

Solution First find the minors of each element in the

second column.

7.5 Finding the Determinant

201341232

8)2)(1()3(23122det

2)2)(1()2(22122det

5)3)(1()2(12131det

32

22

12

M

M

M

Copyright © 2007 Pearson Education, Inc. Slide 7-11

Now, find the cofactor.

The determinant is found by multiplying each cofactor by its corresponding element in the matrix and finding the sum of these products.

7.5 Finding the Determinant

8)8()1()1(2)2()1()1(5)5()1()1(

532

2332

422

2222

312

2112

MAMAMA

23)8)(0()2)(4()5(3

201341232

det 323222221212

AaAaAa

Copyright © 2007 Pearson Education, Inc. Slide 7-12

7.5 Cramer’s Rule for 2 × 2 Systems

• Note: Cramer’s rule does not apply if D = 0.

For the system

where, if possible,

,222

111

cybxacybxa

,andDD

yDDx yx

1 1

2 2

det

x

c bD

c b1 1

2 2

det

y

a cD

a c1 1

2 2

det

z

a bD

a b

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Example Use Cramer’s rule to solve the system.

Analytic Solution By Cramer’s rule,

7.5 Applying Cramer’s Rule to a System with Two Equations

186175

yxyx

. and DyD

DxD yx

11)1)(6()1)(5(1615det

15)7)(1()8)(1(8171det

2)7(6)8(58675det

y

x

D

D

D

Copyright © 2007 Pearson Education, Inc. Slide 7-14

The solution set is

Graphing Calculator SolutionEnter D, Dx, and Dy as matrices A, B, and C, respectively.

7.5 Applying Cramer’s Rule to a System with Two Equations

211

211and

215

215

DD

yDDx yx

., 211

215

Copyright © 2007 Pearson Education, Inc. Slide 7-15

7.5 Cramer’s Rule for a System with Three Equations

For the system

where

,3333

2222

1111

dzcybxadzcybxadzcybxa

,and,,DDz

DD

yDDx zyx

1 1 1 1 1 1

2 2 2 2 2 2

3 3 3 3 3 3

1 1 1 1 1 1

2 2 2 2 2 2

3 3 3 3 3 3

det , det ,

det , and det 0.

x y

z

d b c a d cD d b c D a d c

d b c a d c

a b d a b cD a b d D a b c

a b d a b c