5.5 using determinants solving systems of equations by · pdf file3 4 section solving systems...

SECTION 5.5 • Solving Systems of Equations by Using Determinants 1

OBJECTIVE A To evaluate a determinant

A matrix is a rectangular array of numbers. Eachnumber in a matrix is called an element of the matrix.The matrix at the right, with three rows and fourcolumns, is called a (read “3 by 4”) matrix.

A matrix of m rows and n columns is said to be of order . The matrix above hasorder . The notation refers to the element of a matrix in the ith row and the jthcolumn. For matrix A, , , and .

A square matrix is one that has the same number ofrows as columns. A matrix and a matrixare shown at the right. Associated with every squarematrix is a number called its determinant.

Find the value of the determinant .

The value of the determinant is 10.

For a square matrix whose order is or greater, the value of the determinant is foundby using determinants.

The minor of an element in a determinant is the determinant that isobtained by eliminating the row and column that contain that element.

Find the minor of for the determinant .

The minor of is the determinant created by eliminating the row andcolumn that contain .

Eliminate the row and column as shown:

The minor of is .� 0

�1

8

6 ��3

� 2

0

�1

�3

4

3

4

8

6��3

2 � 2�3

� 2

0

�1

�3

4

3

4

8

6��3

2 � 23 � 3

2 � 23 � 3

� 3

�1

4

2 � � 3 � 2 � 4��1� � 6 � ��4� � 10

� 3

�1

4

2 �

3 � 32 � 2

a13 � 2a31 � 6a23 � �3aij3 � 4

m � n

3 � 4

S E C T I O N

Solving Systems of Equations by Using Determinants5.5

A � �1

0

6

�3

4

�5

2

�3

4

4

2

�1�

��1

5

3

2� �4

5

2

0

�3

1

1

7

4�

Determinant of a 2 � 2 Matrix

The determinant of a matrix is written . The value of this determinant

is given by the formula

� a11

a21

a12

a22� � a11 a22 � a12 a21

� a11

a21

a12

a22��a11

a21

a12

a22�2 � 2

HOW TO • 1

HOW TO • 2

Point of InterestThe word matrix was firstused in a mathematicalcontext in 1850. The root ofthis word is the Latin mater,meaning mother. A matrixwas thought of as an objectfrom which something elseoriginates. The idea was thata determinant, discussedbelow, originated (was born)from a matrix. Today, matricesare one of the most widelyused tools of appliedmathematics.

Take NoteNote that vertical bars areused to represent thedeterminant and that bracketsare used to represent thematrix.

46951_20_5.5online.qxd 1/13/10 10:58 AM

2 CHAPTER 5 • Systems of Linear Equations and Inequalities

For the determinant , find the cofactor of and

of .

Because is in the first row and the second column, and . Thus

. The cofactor of is .

Because is in the second row and the second column, and .

Thus . The cofactor of is .

Note from this example that the cofactor of an element is times the minor of thatelement or 1 times the minor of that element, depending on whether the sum is anodd or an even integer.

The value of a or larger determinant can be found by expanding by cofactors ofany row or any column.

Find the value of the determinant .

We will expand by cofactors of the first row. Any row or column would work.

To illustrate that any row or column can be chosen when expanding by cofactors, we willnow show the evaluation of the same determinant by expanding by cofactors of the sec-ond column.

� 3�2� � 3�4� � 2��4� � 6 � 12 � ��8� � 10

� 3�2 � 0� � 3�4 � 0� � 2��2 � 2�

� �3��1� � 10 �1

2 � � 3�1� � 20 2

2 � � ��2��1� � 21 2

�1 �� 3��1�1�2 � 10 �1

2 � � 3��1�2�2 � 20 2

2 � � ��2��1�3�2 � 21 2

�1 � �210 �3

3

�2

2

�1

2�� 2(4) � 3(2) � 2(�2) � 8 � 6 � 4 � 10

� 2(6 � 2) � 3(2 � 0) � 2(�2 � 0)

� 2(1) � 3

�2

�1

2 � � ��3�(�1) � 10 �1

2 � � 2(1) � 10 3

�2 �� 2(�1)1�1 � 3

�2

�1

2 � � ��3�(�1)1�2 � 10 �1

2 � � 2(�1)1�3 � 10 3

�2 � �210 �3

3

�2

2

�1

2��210 �3

3

�2

2

�1

2�3 � 3

i � j�1

1 � � 30 1

1 ��5��1�i� j � ��1�2�2 � ��1�4 � 1

j � 2i � 2�5

��1� � 20 �4

1 ��2��1�i� j � ��1�1�2 � ��1�3 � �1

j � 2i � 1�2

�5

�2�320 �2

�5

3

1

�4

1�

Cofactor of an Element of a Matrix

The cofactor of an element of a matrix is times the minor of that element, where i is therow number of the element and j is the column number of the element.

��1�i� j

HOW TO • 3

HOW TO • 4

Take NoteThe only difference betweenthe cofactor and the minor ofan element is one of sign.The definition at the right canbe stated with the use ofsymbols as follows: If isthe cofactor and is theminor of the matrix element

, then . Ifis an even number, then

and . Ifis an odd number, then

and .Ci j � �Mi j��1�i � j � �1i � j

Ci j � Mi j��1�i � j � 1i � j

Ci j � ��1�i � jMi jai j

Mi j

Ci j

46951_20_5.5online.qxd 1/13/10 10:58 AM


Note that the value of the determinant is the same whether the first row or the secondcolumn is used to expand by cofactors. Any row or column can be used to evaluate adeterminant by expanding by cofactors.

EXAMPLE • 1 YOU TRY IT • 1

Find the value of.

Solution


Find the value of .

Your solution

� �1

3

�4

�5 �

� 36 �2

�4 � � 3��4� � ��2��6� � �12 � 12 � 0

� 36

�2

�4 �


Find the value of .

Solution

Expand by cofactors of the first row.

The value of the determinant is .

Find the value of .

Your solution

�130 4

1

�2

�2

1

2�

�30

� �30� 12 � 36 � 6� �2��6� � 3�12� � 1��6�� 2��6 � 0� � 3�12 � 0� � 1��8 � 2�

� �2 � �2

�2

0

3 � � 3 � 41 0

3 � � 1 � 41 �2

�2 ��2

4

1

3

�2

�2

1

0

3��2

4

1

3

�2

�2

1

0

3�

Solutions on p. S1


Find the value of .

Solution


Find the value of .

Your solution

� 31�2

�2

4

1

0

2

3�

�7

� �7� 4 � 11� 2�2� � 1��11�� 0 � ��2��4 � 2� � 1��3 � 8�

� 0 � 4

�3

1

4 � � ��2� � 12 1

4 � � 1 � 12 4

�3 ��012 �2

4

�3

1

1

4��012 �2

4

�3

1

1

4�

46951_20_5.5online.qxd 1/13/10 10:58 AM


OBJECTIVE B To solve a system of equations by using Cramer’s Rule

The connection between determinants and systems of equations can be understood bysolving a general system of linear equations.

Solve: (1) (2)

Eliminate y. Multiply Equation (1) by and Equation (2) by .

Add the equations.

The denominator is the deter-minant of the coefficients of x and y. This iscalled the coefficient determinant.

The numerator is the determinantobtained by replacing the first column in the coefficient determinant by the constants

and . This is called a numerator deter-minant.

By following a similar procedure and eliminating x, it is also possible to express the y-component of the solution in determinant form. These results are summarized inCramer’s Rule.

Solve by using Cramer’s Rule:

• Find the value of the coefficient determinant.

,

, • Use Cramer’s Rule to write the solution.

The solution is , .7

19�11

19

y �Dy

D�

7

19x �

Dx

D�

11

19

Dy � � 32 1

3 � � 7Dx � � 13 �2

5 � � 11

D � � 32 �2

5 � � 19

2x � 5y � 33x � 2y � 1

c2c1

c1 b2 � c2 b1

a1 b2 � a2 b1

x �c1 b2 � c2 b1

a1 b2 � a2 b1

�a1 b2 � a2 b1�x � c1 b2 � c2 b1

a1 b2 x � a2 b1 x � c1 b2 � c2 b1

�a2 b1 x � b1 b2 y � �c2 b1

a1 b2 x � b1 b2 y � c1 b2

�b1b2

a2 x � b2 y � c2

a1 x � b1 y � c1

• times Equation (1)• times Equation (2)�b1

b2

• Solve for x, assuming .a1b2 � a2b1 � 0

• Find the value of each of thenumerator determinants.

coefficients of xcoefficients of y

a1 b2 � a2 b1 � � a1

a2

b1

b2�

constants ofthe equations

c1 b2 � c2 b1 � � c1

c2

b1

b2�

Cramer’s Rule

The solution of the system of equations is given by and ,

where , and .D � 0Dy � � a1

a2

c1

c2�D � � a1

a2

b1

b2� , Dx � � c1

c2

b1

b2� ,

y �Dy

Dx �

Dx

Da1x � b1y � c1

a2x � b2 y � c2

Point of InterestCramer’s Rule is named afterGabriel Cramer, who used itin a book he published in1750. However, this rule wasalso published in 1683 by theJapanese mathematician SekiKowa. That publicationoccurred seven years beforeCramer’s birth.

HOW TO • 5

46951_20_5.5online.qxd 1/13/10 10:58 AM


A procedure similar to that followed for two equations in two variables can be used toextend Cramer’s Rule to three equations in three variables.

Solve by using Cramer’s Rule:

Find the value of the coefficient determinant.

Find the value of each of the numerator determinants.

Use Cramer’s Rule to write the solution.

The solution is , , .30

235

23��1

23

x �Dx

D�

�1

23, y �

Dy

D�

5

23, z �

Dz

D�

30

23

� 30 � 2�14� � 1�10� � 1��8�

Dz � �213 �1

3

1

1

�2

4� � 2 � 31 �2

4 � � ��1� � 13 �2

4 � � 1 � 13 3

1 � � 5 � 2�2� � 1�9� � 1�10�

Dy � �213 1

�2

4

1

�2

3� � 2 � �2

4

�2

3 � � 1 � 13 �2

3 � � 1 � 13 �2

4 � � �1 � 1�11� � 1�2� � 1��14�

Dx � � 1

�2

4

�1

3

1

1

�2

3� � 1 � 31 �2

3 � � ��1� � �2

4

�2

3 � � 1 � �2

4

3

1 �

� 23 � 2�11� � 1�9� � 1��8�

D � �213 �1

3

1

1

�2

3� � 2 � 31 �2

3 � � ��1� � 13 �2

3 � � 1 � 13 3

1 �

3x � y � 3z � 4 x � 3y � 2z � �2 2x � y � z � 1

Cramer’s Rule for a System of Three Equations in Three Variables

The solution of the system of equations

is given by where

.Dz � �a1

a2

a3

b1

b2

b3

d1

d2

d3�, and D � 0Dx � �d1

d2

d3

b1

b2

b3

c1

c2

c3�, Dy � �a1

a2

a3

d1

d2

d3

c1

c2

c3�,D � �a1

a2

a3

b1

b2

b3

c1

c2

c3�,x �

Dx

D, y �

Dy

D, and z �

Dz

D,

a1x � b1y � c1z � d1

a2x � b2y � c2z � d2

a3x � b3y � c3z � d3

HOW TO • 6

46951_20_5.5online.qxd 1/13/10 10:58 AM


EXAMPLE • 4 YOU TRY IT • 4Solve by using Cramer’s Rule.

Solution

Because , is undefined. Therefore, the

system is dependent or inconsistent.

Solve by using Cramer’s Rule.

Your solution

6x � 2y � 5 3x � y � 4

Dx

DD � 0

D � � 64 �9

�6 � � 0

4x � 6y � 46x � 9y � 5

Solutions on p. S1

EXAMPLE • 5 YOU TRY IT • 5Solve by using Cramer’s Rule.

Solution

The solution is (1, 0, 2).

Solve by using Cramer’s Rule.

Your solution

x � 3y � z � �2 3x � 2y � z � 3 2x � y � z � �1

z �Dz

D�

56

28� 2

y �Dy

D�

0

28� 0

x �Dx

D�

28

28� 1

Dz � �312 �1

2

3

5

�3

4� � 56

Dy � �312 5

�3

4

1

�2

1� � 0,

Dx � � 5

�3

4

�1

2

3

1

�2

1� � 28,

D � �312 �1

2

3

1

�2

1� � 28,

2x � 3y � z � 4 x � 2y � 2z � �3

3x � y � z � 5

46951_20_5.5online.qxd 1/13/10 10:58 AM


5.5 EXERCISES

OBJECTIVE A To evaluate a determinant

1. How do you find the value of the determinant associated with a matrix?

2. What is the cofactor of a given element in a matrix?

For Exercises 3 to 14, evaluate the determinant.

3. 4. 5. 6.

7. 8. 9. 10.

11. 12. 13. 14.

15. What is the value of a determinant for which one row is all zeros?

16. What is the value of a determinant for which all the elements are the same number?

For Exercises 17 to 34, solve by using Cramer’s Rule.

17. 18. 19. 20.

21. 22. 23. 24.5x � 4y � 9 6x � 2y � 1 15x � 8y � �21 6x � 12y � �57x � 3y � 4 2x � 5y � 6 5x � 4y � 3 2x � 3y � 4

3x � 4y � 11 3x � 7y � 5 2x � 5y � 11 5x � 3y � 35x � 2y � �5 x � 4y � 8 3x � 7y � 15 2x � 5y � 26

� 3

4

�1

6

�1

�2

�3

6

3�� 4

�2

2

2

1

1

6

1

3��432 5

�1

1

�2

5

4��303 �1

1

2

2

2

�2��423 1

�2

1

3

1

2��131 �1

2

0

2

1

4�� 51 �10

�2 �� 32 6

4 �

� �3

1

5

7 �� 6

�3

�2

4 �� 5

�1

1

2 �� 23 �1

4 �

2 � 2

OBJECTIVE B To solve a system of equations by using Cramer’s Rule

46951_20_5.5online.qxd 1/13/10 10:58 AM


25. 26. 27. 28.

29. 30. 31.

32. 33. 34.

35. Can Cramer’s Rule be used to solve a dependent system of equations?

36. Suppose a system of linear equations in two variables has , , and . Is the system of equa-tions independent, dependent, or inconsistent?

Applying the Concepts

37. Determine whether the following statements are always true, sometimes true, ornever true.a. The determinant of a matrix is a positive number.b. A determinant can be evaluated by expanding about any row or column of the

matrix.c. Cramer’s Rule can be used to solve a system of linear equations in three vari-

ables.

38. Show that .

39. Surveying Surveyors use a formula to find the area of a plot of land. The survey-or’s area formula states that if the vertices , of a simplepolygon are listed counterclockwise around the perimeter, then the area of the poly-gon is

Use the surveyor’s area formula to find the area of the polygon with vertices , (26, 6), (18, 21), (16, 10), and (1, 11). Measurements are given in feet.�9, �3�

A �1

2� � x1

y1

x2

y2� � � x2

y2

x3

y3� � � x3

y3

x4

y4� � � � � � � xn

yn

x1

y1� �

�x2, y2�, . . .,�xn, yn��x1, y1�

� ac b

d � � � � ca d

b �

D � 0Dy � 0Dx � 0

3x � 9y � 6z � �3 2x � y � 3z � 2 3x � 4y � 2z � 9 2x � y � 2z � 3 3x � 4y � 2z � 1 2x � 3y � z � 5 x � 3y � 2z � 1 4x � 2y � 6z � 1 x � 2y � 3z � 8

2x � 2y � z � �3 3x � y � z � 0 3x � 2y � 2z � 5 x � 4y � 2z � �12 2x � 3y � 2z � �6 x � 4y � 4z � 5

3x � y � z � 11 3x � 2y � z � 2 2x � y � 3z � 9

2x � 2y � 5 3x � 7y � �33x � 2y � 4 4x � 6y � 9 8x � 7y � �3 2x � 5y � �29x � 6y � 7 �2x � 3y � 7

© S

penc

er G

rant

/Pho

toEd

it, In

c.

46951_20_5.5online.qxd 1/13/10 10:58 AM

Solutions to You Try It S1

SOLUTIONS TO CHAPTER 5 “YOU TRY IT”

SECTION 5.5

You Try It 1


You Try It 2 Expand by cofactors of the first row.


You Try It 3


You Try It 4

Because , is undefined.

Therefore, the system of equations is dependent or inconsistent. That is, thesystem is not independent.

You Try It 5

The solution is . �2�3

7 , �

1

7 ,

z �Dz

D�

�42

21� �2

y �Dy

D�

�3

21� �

1

7

x �Dx

D�

9

21�

3

7

Dz � �231 �1

2

3

�1

3

�2� � �42

Dy � �231 �1

3

�2

1

�1

1� � �3

Dx � ��1

3

�2

�1

2

3

1

�1

1� � 9

D � �231 �1

2

3

1

�1

1� � 21

x � 3y � z � �23x � 2y � z � 3

2x � y � z � �1

Dx

DD � 0

D � � 36 �1

�2 � � 0

6x � 2y � 5 3x � y � 4

� 44� 30 � 14� 3�10� � 2�7�� 3�12 � 2� � 2�3 � 4� � 0

� 3 � 41 2

3 � � ��2� � 1

�2

2

3 � � 0 � 1

�2

4

1 �� 3

1

�2

�2

4

1

0

2

3��8

� �8� 4 � 24 � 12� 1�2 � 2� � 4�6 � 0� � 2��6 � 0�

� 1 � 1

�2

1

2 � � 4 � 30 1

2 � � ��2� � 30 1

�2 ��130 4

1

�2

�2

1

2� � 5 � 12 � 17

� �1

3

�4

�5 � � �1��5� � ��4��3�

46951_20_5.5online.qxd 1/13/10 10:58 AM Page S1

Answers to Selected Exercises A1

ANSWERS TO CHAPTER 5 SELECTED EXERCISES

SECTION 5.5

3. 11 5. 18 7. 0 9. 15 11. 13. 0 15. 0 17. 19. 21. , 23. , 1

25. Not independent 27. 29. 31. 33. Not independent 35. No 37. a. Sometimes

true b. Always true c. Sometimes true 39. The area is .239 ft2

�2, �2, 3��1, �1, 2��1, 0�

1

2�17

21

11

14��4, �1��3, �4��30

46951_20_5.5online.qxd 1/13/10 10:58 AM Page A1

5.5 using determinants solving systems of equations by · pdf file3 4 section solving systems...

Documents