5.3 - 1 10 TH EDITION LIAL HORNSBY SCHNEIDER COLLEGE ALGEBRA.

5.3 - 1

10TH EDITION

LIAL

HORNSBY

SCHNEIDER

COLLEGE ALGEBRA

5.3 - 25.3 - 2

5.3Determinant Solution of Linear EquationsDeterminantsCofactorsEvaluating n n DeterminantsCramer’s Rule

5.3 - 35.3 - 3

Determinants

Every n n matrix A is associated with a real number called the determinant of A, written A. The determinant of a 2 2 matrix is defined as follows.

5.3 - 45.3 - 4

Determinant of a 2 2 Matrix

If A = 11 12

21 22

, thena a

a a

11 1211 22 21 12

21 22

.a a

A a a a aa a

5.3 - 55.3 - 5

Note Matrices are enclosed with square brackets, while determinantsare denoted with vertical bars. A matrix is an array of numbers, but its determinantis a single number.

5.3 - 65.3 - 6

Determinants

The arrows in the following diagram will remind you which products to find when evaluating a 2 2 determinant.

5.3 - 75.3 - 7

Example 1 EVALUATING A 2 2 DETERMINANT

Let A = 3 4

.6 8

Find A.

Use the definition with

Solution

11 12 21 223, 4, 6, 8.a a a a

3 8 6 4A

a11 a22 a21 a12

24 24 48

5.3 - 85.3 - 8

Determinant of a 3 3 Matrix

If A =

11 12 13

21 22 23

31 32 33

, then

a a a

a a a

a a a

11 12 13

21 22 23 11 22 33 12 23 31 13 21 32

31 32 33

( )

a a a

A a a a a a a a a a a a a

a a a

31 22 13 32 23 11 33 21 12( ).a a a a a a a a a

5.3 - 95.3 - 9

Evaluating

The terms on the right side of the equation in the definition of A can be rearranged to get

12 13

22 23

11

21 11 2122 33 32 23 12 33 32 13

32 3331

( ) ( )

a a

A a a a a a a a a a a

a a

a

a a a

a

12 23 2 331 2 1( ).a a a aa

Each quantity in parentheses represents the determinant of a 2 2 matrix that is the part of the matrix remaining when the row and column of themultiplier are eliminated, as shown in the next slide.

5.3 - 105.3 - 10

Evaluating

22 33 3 31 21 2( )a a a aa

12 33 3 31 22 1( )a a a aa

12 23 2 31 23 1( )a a a aa

11 12 13

21 22 23

32 31 33

a a a

a

a

a a

a a

11

21 2

1

2

2 13

32 3

23

3 31

a

a a a

a

a a

a a

11

21

31 32

12 13

22

33

23

a

a

a a

a a

a

a a

5.3 - 115.3 - 11

Cofactors

The determinant of each 2 2 matrix above is called the minor of the associated element in the 3 3 matrix. The symbol represents Mij,the minor that results when row i and column j are eliminated. The following table in the next slide gives some of the minors from the previous matrix.

5.3 - 125.3 - 12

Cofactors

Element Minor Element Minor

a11a22

a21a23

a31a33

22 2311

32 33

a aM

a a

12 1321

32 33

a aM

a a

12 1331

22 23

a aM

a a

11 1322

31 33

a aM

a a

11 1223

31 32

a aM

a a

11 1233

21 22

a aM

a a

5.3 - 135.3 - 13

Cofactors

In a 4 4 matrix, the minors are determinants of matrices. Similarly, an n n matrix has minors that are determinants of matrices. To find the determinant of a 3 3 or larger matrix, first choose any row or column. Then the minor of each element in that row or column must be multiplied by +1 or – 1, depending on whether the sum of the row number and column number is even or odd. The product of a minor and the number +1 or – 1 is called a cofactor.

5.3 - 145.3 - 14

Cofactor

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

( 1) .i jij ijA M

5.3 - 155.3 - 15

Example 2 FINDING COFACTORS OF ELEMENTS

Find the cofactor of each of the following elements of the matrix

3

6 4

8

2

9 .

1 2 0

a. 6Solution Since 6 is in the first row, first column of the matrix, i = 1 and j = 1 so 11

9 36.

2 0M

The cofactor is 1 1( 1) ( 6) 1( 6) 6.

5.3 - 165.3 - 16



3

6 4

8

2

9 .

1 2 0

b. 3Solution

Here i = 2 and j = 3 so, 23

6 210.

1 2M

The cofactor is 2 3( 1) (10) 1(10) 10.

5.3 - 175.3 - 17



3

6 4

8

2

9 .

1 2 0

c. 8Solution

We have, i = 2 and j = 1 so, 21

2 48.

2 0M

The cofactor is 2 1( 1) ( 8) 1( 8) 8.

5.3 - 185.3 - 18

Finding the Determinant of a Matrix

Multiply each element in any row or column of the matrix by its cofactor.The sum of these products gives the value of the determinant.

5.3 - 195.3 - 19


Evaluate

2 3 2

1 4 3 ,

1 0 2

expanding by the second column.

Solution

12

1 31(2) ( 1) 3)

1 25(M

22

2 22(2) ( 1)( 2)

1 22M

32

2 22( 3) ( 1)( 2)

1 38M

Use parentheses, & keep track of all negative signs to avoid errors.

5.3 - 205.3 - 20


Now find the cofactor of each element of these minors.

121 2

123( 1) ( 1) ( 5) ( 5) 51MA

222

4222 ( 1) ( 1) (2) 1 2 2MA

323 2

325( 1) ( 1) ( 8) ( 8) 81MA

5.3 - 215.3 - 21


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

12 12 2 32 2 222 3

2 2

1

1 2

4 3

3

0

a A aA Aa

(5) ( )2 (8043 )

15 ( 8) 0 23

5.3 - 225.3 - 22

Cramer’s Rule

Determinants can be used to solve a linear system in the form

1 1 1a x b y c (1)

2 2 2a x b y c (2)

by elimination as follows.

1 1

2 2 21 1 1

2 2 2 1

1 2 2 1 1 2 2 1( )

a x b y c

a x b y ca b a b x c b c

b b b

b

b

b b

Multiply (1) by b2 .

Multiply (2) by – b1 .

Add.

5.3 - 235.3 - 23

Cramer’s Rule

2 2 21 1 1

2 2 2

1 2 2 1

1

1

1

1

1

2 2( )

a x b y c

a x b y

a a a

ca b a b y a c a c

a a a

1 2 2 11 2 2 1

1 2 2 1

, if 0.a c a c

y a b a ba b a b

Multiply (1) by – a2 .

Multiply (2) by a1 .

Add.

Similarly,

5.3 - 245.3 - 24

Cramer’s Rule

Both numerators and the common denominator of these values for x and y can be written as determinants, since

1 1 1 11 2 2 1 1 2 2 1

2 2 2 2

1 11 2 2 1

2 2

, ,

and .

c b a cc b c b a c a c

c b a c

a ba b a b

a b

5.3 - 255.3 - 25

Cramer’s Rule

Using these determinants, the solutions for x and y become

1 1 1 1

1 12 2 2 2

1 1 1 1 2 2

2 2 2 2

and , if 0.

c b a c

a bc b a cx y

a b a b a b

a b a b

5.3 - 265.3 - 26

Cramer’s Rule

We denote the three determinants in the solution as

1 1 1 1 1 1

2 2 2 2 2 2

, , and .x y

a b c b a cD D D

a b c b a c

5.3 - 275.3 - 27

Note The elements of D are the four coefficients of the variables in thegiven system. The elements of Dx are obtained by replacing the coefficientsof x in D by the respective constants, and the elements of Dy are obtained byreplacing the coefficients of y in D by the respective constants.

5.3 - 285.3 - 28

Cramer’s Rule for Two Equations in Two VariablesGiven the system

1 1 1a x b y c

2 2 2a x b y c

if then the system has the unique solution

and ,yxDD

x yD D

where 1 1 1 1 1 1

2 2 2 2 2 2

, , and .x y

a b c b a cD D D

a b c b a c

5.3 - 295.3 - 29

Caution As indicated in the preceding box, Cramer’s rule does not apply if D = 0. When D = 0 the system is inconsistent or has infinitely many solutions. For this reason, evaluate D first.

5.3 - 305.3 - 30

Example 4 APPLYING CRAMER’S RULE TO A 2 2 SYSTEM

Use Cramer’s rule to solve the system5 7 1

6 8 1

x y

x y

Solution

By Cramer’s rule, and Find D first, since if D = 0, Cramer’s rule does not apply. If D ≠ 0, then find Dx and Dy.

xDx

D .yD

yD

5 75(8) 6(7)

62

8D

1 71(8)

1 851(7) 1xD

5 15(1) 16( 1) 1

6 1yD

5.3 - 315.3 - 31


By Cramer’s rule,

15 11 and .

2112

52

12

x yDDx y

D D

The solution set is as can be verified by substituting in the given system.

15 11,

2 2

5.3 - 325.3 - 32

General form of Cramer’s RuleLet an n n system have linear equations of the form 1 1 2 2 3 3 .n na x a x a x a x b Define D as the determinant of the n n matrix of all coefficients of the variables. Define Dx1 as the determinant obtained from D by replacing the entries in column 1 of D with the constants of the system. Define Dxi as the determinant obtained from D by replacing the entries in column i with the constants of the system. If D 0, the unique solution of the system is

31 21 2 3, , , , .xx x xn

n

DD D Dx x x x

D D D D

5.3 - 335.3 - 33


Use Cramer’s rule to solve the system.2 0x y z

2 5 0x y z 2 3 4 0x y z

Solution 2x y z

2 5x y z

2 3 4x y z

Rewrite each equation in the form ax + by + cz + = k.

5.3 - 345.3 - 34


Verify that the required determinants are

1 1 1

2 1 1 3,

1 2 3

D

2 1 1

5 1 1 7,

4 2 3xD

1 2 1

2 5 1 22,

1 4 3yD

1 1 2

2 1 5 21.

1 2 4zD

5.3 - 355.3 - 35


Thus,

7 7 22 22, ,

3 3 3 3yx

DDx y

D D

and21

7,3

zDz

D

so the solution set is 7 22, , 7 .

3 3

5.3 - 365.3 - 36

Caution As shown in Example 5, each equation in the system must bewritten in the form ax + by + cz + = k before using Cramer’s rule.

5.3 - 375.3 - 37

Example 6 SHOWING THAT CRAMER’S RULE DOES NOT APPLY

Show that Cramer’s rule does not apply to the following system. 2 3 4 10x y z

6 9 12 24x y z 2 3 5x y z

We need to show that D = 0. Expanding about column 1 gives

2 3 49 12 3 4 3 4

6 9 12 2 6 12 3 2 3 9 12

1 2 3

D

2(3) 6(1) 1(0 0) .

Since D = 0, Cramer’s rule does not apply.

Solution

5.3 - 385.3 - 38

Note When D = 0, the system is either inconsistent or has infinitely many solutions. Use the elimination method to tell which is the case. Verify that the system in Example 6 is inconsistent, so the solution set is ø.

5.3 - 1 10 TH EDITION LIAL HORNSBY SCHNEIDER COLLEGE ALGEBRA.

Documents