8.5

856 Chapter 8 Matrices and Determinants

Determinants and Cramer’s Rule

A s cyberspace absorbs more andmore of our work, play, shopping,

and socializing, where will it all end?Which activities will still be offline in2025?

Our technologically transformedlives can be traced back to the Englishinventor Charles Babbage (1792–1871).Babbage knew of a method for

solving linear systems called Cramer’s rule, in honor of the Swiss geometer GabrielCramer (1704–1752). Cramer’s rule was simple, but involved numerousmultiplications for large systems. Babbage designed a machine, called the“difference engine,” that consisted of toothed wheels on shafts for performingthese multiplications. Despite the fact that only one-seventh of the functions everworked, Babbage’s invention demonstrated how complex calculations could behandled mechanically. In 1944, scientists at IBM used the lessons of the differenceengine to create the world’s first computer.

Those who invented computers hoped to relegate the drudgery of repeatedcomputation to a machine. In this section, we look at a method for solving linearsystems that played a critical role in this process. The method uses real numbers,called determinants, that are associated with arrays of numbers. As with matrixmethods, solutions are obtained by writing down the coefficients and constants of alinear system and performing operations with them.

The Determinant of a Matrix

Associated with every square matrix is a real number, called its determinant. Thedeterminant for a square matrix is defined as follows:2 * 2

2 : 2

Objectives

� Evaluate a second-orderdeterminant.

� Solve a system of linearequations in two variablesusing Cramer’s rule.

� Evaluate a third-orderdeterminant.

� Solve a system of linearequations in three variablesusing Cramer’s rule.

� Use determinants to identifyinconsistent systems andsystems with dependentequations.

� Evaluate higher-orderdeterminants.

S e c t i o n

90. If find

91. Find values of for which the following matrix is not invertible:

Group Exercise92. Each person in the group should work with one partner.

Send a coded word or message to each other by giving yourpartner the coded matrix and the coding matrix that youselected. Once messages are sent, each person should decodethe message received.

B 1 a + 1a - 2 4

R .

a

1A-12-1.A = B3 5

2 4R ,

Preview ExercisesExercises 93–95 will help you prepare for the material covered inthe next section. Simplify the expression in each exercise.

93.

94.

95. 21-30 - 1-322 - 316 - 92 + 1-1211 - 152

21-52 - 11-42

51-52 - 61-42

21-52 - 1-32142

A portion of Charles Babbage’s unrealizedDifference Engine

� Evaluate a second-orderdeterminant.

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 856

DiscoveryWrite and then evaluate threedeterminants, one whose value ispositive, one whose value is negative,and one whose value is 0.

Section 8.5 Determinants and Cramer’s Rule 857

Study TipTo evaluate a second-orderdeterminant, find the difference of theproduct of the two diagonals.

`a1

a1b2

b1

a2a2b1b2

` = -

Definition of the Determinant of a Matrix

The determinant of the matrix is denoted by and is defined by

We also say that the value of the second-order determinant is

a1b2 - a2b1 .

`a1 b1

a2 b2`

`a1 b1

a2 b2` = a1b2 - a2b1 .

`a1 b1

a2 b2`Ba1 b1

a2 b2R2 : 2

Example 1 illustrates that the determinant of a matrix may be positive ornegative. A determinant can also have 0 as its value.

Evaluating the Determinant of a Matrix

Evaluate the determinant of each of the following matrices:

a. b.

Solution We multiply and subtract as indicated.

a. `5 67 3

` = 5 # 3 - 7 # 6 = 15 - 42 = -27

B 2 4-3 -5

R .B5 67 3

R2 : 2EXAMPLE 1

� Solve a system of linearequations in two variables using Cramer’s rule.

x =

c1b2 - c2b1

a1b2 - a2b1.

1a1b2 - a2b12x = c1b2 - c2b1

b a1b2x + b1b2y = c1b2

-a2b1x - b1b2y = -c2b1

Because

`c1 b1

c2 b2` = c1b2 - c2b1 and `

a1 b1

a2 b2` = a1b2 - a2b1 ,

The value of the second-order determinant is -27.

b. `2 4

-3 -5` = 21-52 - 1-32142 = -10 + 12 = 2

The value of the second-order determinant is 2.

Check Point 1 Evaluate the determinant of each of the following matrices:

a. b.

Solving Systems of Linear Equations in Two Variables Using DeterminantsDeterminants can be used to solve a linear system in two variables. In general, sucha system appears as

Let’s first solve this system for using the addition method. We can solve for byeliminating from the equations. Multiply the first equation by and the secondequation by Then add the two equations:

ba1x + b1y = c1

a2x + b2y = c2

-b1 .b2y

xx

ba1x + b1y = c1

a2x + b2y = c2 .

B 4 3-5 -8

R .B10 96 5

R

Add:

Multiply by -b1 . "

Multiply by b2 . "

"

"

" "

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 857

858 Chapter 8 Matrices and Determinants

we can express our answer for as the quotient of two determinants:

Similarly, we could use the addition method to solve our system for again expressing as the quotient of two determinants. This method of using determinants to solve thelinear system, called Cramer’s rule, is summarized in the box.

yy,

x =

c1b2 - c2b1

a1b2 - a2b1=

`c1 b1

c2 b2`

`a1 b1

a2 b2`

.

x

Solving a Linear System in Two Variables Using DeterminantsCramer’s RuleIf

then

where

`a1 b1

a2 b2` Z 0.

x =

`c1 b1

c2 b2`

`a1 b1

a2 b2`

and y =

`a1 c1

a2 c2`

`a1 b1

a2 b2`

,

b a1x + b1y = c1

a2x + b2y = c2,

Here are some helpful tips when solving

using determinants:

1. Three different determinants are used to find and The determinants in thedenominators for and are identical. The determinants in the numeratorsfor and differ. In abbreviated notation, we write

2. The elements of the determinant in the denominator, are the coefficients ofthe variables in the system.

3. the determinant in the numerator of is obtained by replacing thein and with the constants on the right sides of the

equations, and

D = `a1 b1

a2 b2` and Dx = `

c1 b1

c2 b2`

c2 .c1

a2 ,D, a1x-coefficients,x,Dx ,

D = `a1 b1

a2 b2`

D,

x =

Dx

D and y =

Dy

D, where D Z 0.

yxyx

y.x

ba1x + b1y = c1

a2x + b2y = c2

Replace the column with and with the constants and to get Dx .c2c 1

a2a1

4. the determinant in the numerator for is obtained by replacing thein and with the constants on the right sides of the

equations, and c2 .c1

b2 ,D, b1y-coefficients,y,Dy ,

D = `a1 b1

a2 b2` and Dy = `

a1 c1

a2 c2`

Replace the column with and with the constants and to get Dy .c2c1

b2b1

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 858

Section 8.5 Determinants and Cramer’s Rule 859

Using Cramer’s Rule to Solve a Linear System

Use Cramer’s rule to solve the system:

Solution Because

we will set up and evaluate the three determinants and

1. the determinant in both denominators, consists of the - and

Because this determinant is not zero, we continue to use Cramer’s rule to solvethe system.

2. the determinant in the numerator for is obtained by replacing thein 5 and 6, by the constants on the right sides of the equations,

2 and 1.

3. the determinant in the numerator for is obtained by replacing thein and by the constants on the right sides of the

equations, 2 and 1.

4. Thus,

As always, the solution (6, 7) can be checked by substituting these values intothe original equations. The solution set is

Check Point 2 Use Cramer’s rule to solve the system:

The Determinant of a MatrixAssociated with every square matrix is a real number called its determinant. Thedeterminant for a matrix is defined as follows:3 * 3

3 : 3

e5x + 4y = 123x - 6y = 24.

516, 726.

x =Dx

D=

-6-1

= 6 and y =Dy

D=

-7-1

= 7.

Dy = ` 5 26 1

` = 152112 - 162122 = 5 - 12 = -7

-5,D, -4y-coefficientsy,Dy ,

Dx = ` 2 -41 -5

` = 1221-52 - 1121-42 = -10 + 4 = -6

D,x-coefficientsx,Dx ,

D = ` 5 -46 -5

` = 1521-52 - 1621-42 = -25 + 24 = -1

y-coefficients.xD,

Dy .D, Dx ,

x =Dx

Dand y =

Dy

D,

b5x - 4y = 26x - 5y = 1.

EXAMPLE 2

Definition of a Third-Order Determinant

3 a1 b1 c1

a2 b2 c2

a3 b3 c3

3 = a1b2c3 + b1c2a3 + c1a2b3 - a3b2c1 - b3c2a1 - c3a2b1

3 Evaluate a third-orderdeterminant.

M09_BLIT59845_04_SE_C08.QXD 7/9/10 9:50 AM Page 859

860 Chapter 8 Matrices and Determinants

The six terms and the three factors in each term in this complicated evaluation formula,, can be rearranged, and then

we can apply the distributive property.We obtain

You can evaluate each of the second-order determinants and obtain the threeexpressions in parentheses in the second step.

In summary, we now have arranged the definition of a third-order determinantas follows:

= a1 `b2 c2

b3 c3` - a2 `

b1 c1

b3 c3` + a3 `

b1 c1

b2 c2` .

= a11b2c3 - b3c22 - a21b1c3 - b3c12 + a31b1c2 - b2c12

a1b2c3 - a1b3c2 - a2b1c3 + a2b3c1 + a3b1c2 - a3b2c1

a1b2c3 + b1c2a3 + c1a2b3 - a3b2c1 - b3c2a1 - c3a2b1

Definition of the Determinant of a MatrixA third-order determinant is defined by

Subtract. Add.

Each a on the right comesfrom the first column.

a1

a2

a3

b1

b2

b3

c1

c2

c3

b2

b3

c2

c33 3 2 2 b1

b3

c1

c3

22 b1

b2

c1

c2

22=a1 – a2 +a3 .

3 : 3

Evaluating the Determinant of a Matrix

1. Each of the three terms in the definition contains two factors—a numericalfactor and a second-order determinant.

2. The numerical factor in each term is an element from the first column of thethird-order determinant.

3. The minus sign precedes the second term.

4. The second-order determinant that appears in each term is obtained bycrossing out the row and the column containing the numerical factor.

The minor of an element is the determinant that remains after deleting therow and column of that element. For this reason, we call this methodexpansion by minors.

3 a1 b1 c1

a2 b2 c2

a3 b3 c3

a1 b1 c1

a2 b2 c2

a3 b3 c3

a1 b1 c1

a2 b2 c2

a3 b3 c3

3a1 `

b2 c2

b3 c3` - a2 `

b1 c1

b3 c3` + a3 `

b1 c1

b2 c2`

3 3 3 3

3 : 3

Here are some tips that may be helpful when evaluating the determinant of amatrix:3 * 3

Evaluating the Determinant of a Matrix

Evaluate the determinant of the following matrix:

C 4 1 0-9 3 4-3 8 1

S .

3 : 3EXAMPLE 3

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Section 8.5 Determinants and Cramer’s Rule 861

Solution We know that each of the three terms in the determinant contains anumerical factor and a second-order determinant. The numerical factors arefrom the first column of the given matrix. They are highlighted in the followingmatrix:

We find the minor for each numerical factor by deleting the row and column of thatelement:

Now we have three numerical factors, 4, and and three second-orderdeterminants. We multiply each numerical factor by its second-order determinantto find the three terms of the third-order determinant:

Based on the preceding definition, we subtract the second term from the first termand add the third term:

Don't forget tosupply the minus sign.

=4 -(–9)33 2 23

8

4

–9

–3

1

3

8

0

4

1

4

12 21

80

1– 3 2 21

30

4

4 `3 48 1

` , -9 `1 08 1

` , -3 `1 03 4

` .

-3,-9,

The minor for

C S4

–9

–3

1

3

8

0

4

1

4 is .38

41� �

The minor for

−9 is .18

01� �

C S4

–9

–3

1

3

8

0

4

1

C S4

–9

–3

1

3

8

0

4

1

The minor for

−3 is .13

04� �

C 4 1 0-9 3 4-3 8 1

S .

Begin by evaluating the three second-order determinants.

Multiply within parentheses.

Subtract within parentheses.

Multiply.Add and subtract as indicated.

Check Point 3 Evaluate the determinant of the following matrix:

The six terms in the definition of a third-order determinant can be rearrangedand factored in a variety of ways.Thus, it is possible to expand a determinant by minorsabout any row or any column. Minus signs must be supplied preceding any elementappearing in a position where the sum of its row and its column is an odd number. Forexample, expanding about the elements in column 2 gives us

Minus sign is supplied becauseb1 appears in row 1 and column 2;

1 + 2 = 3, an odd number.

Minus sign is supplied becauseb3 appears in row 3 and column 2;

3 + 2 = 5, an odd number.

a1

a2

a3

b1

b2

b3

c1

c2

c3

3 3 a2

a3

c2

c3

2 2=–b1 +b2

a1

a3

c1

c3

2 2 a1

a2

c1

c2

2 2-b3 .

C 2 1 7-5 6 0-4 3 1

S .

= -119 = -116 + 9 - 12

= 41-292 + 9112 - 3142

= 413 - 322 + 911 - 02 - 314 - 02

= 413 # 1 - 8 # 42 + 911 # 1 - 8 # 02 - 311 # 4 - 3 # 02

TechnologyA graphing utility can be used toevaluate the determinant of a matrix.Enter the matrix and call it Thenuse the determinant command. Thescreen below verifies our result inExample 3.

A.

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 861

862 Chapter 8 Matrices and Determinants

Expanding by minors about column 3, we obtain

When evaluating a determinant using expansion by minors, you canexpand about any row or column. To simplify the arithmetic, if a row or columncontains one or more 0s, expand about that row or column.

Evaluating a Third-Order Determinant

Evaluate:

Solution Note that the last column has two 0s. We will expand the determinantabout the elements in that column.

= 0 - 0 + 2391-32 - 1-2254

3 9 5 0-2 -3 0

1 4 2

3 = 0 `-2 -3

1 4` - 0 `

9 51 4

` + 2 `9 5

-2 -3`

3 9 5 0-2 -3 0

1 4 2

3 .

EXAMPLE 4

3 * 3

Minus sign must be supplied becausec2 appears in row 2 and column 3;

2 + 3 = 5, an odd number.

a1

a2

a3

b1

b2

b3

c1

c2

c3

3 3 a2

a3

b2

b3

2 2=c1 -c2

a1

a3

b1

b3

2 2 a1

a2

b1

b2

2 2+c3 .

Study TipKeep in mind that you can expand adeterminant by minors about any rowor column. Use alternating plus andminus signs to precede the numericalfactors of the minors according to thefollowing sign array:

3 + - +

- + -

+ - +

3 .

Check Point 4 Evaluate:

Solving Systems of Linear Equations in Three Variables Using DeterminantsCramer’s rule can be applied to solving systems of linear equations in three variables.The determinants in the numerator and denominator of all variables are third-orderdeterminants.

3 6 4 0-3 -5 3

1 2 0

3 .

= -34

= 21-172

= 21-27 + 102

� Solve a system of linearequations in three variablesusing Cramer’s rule.

Solving Three Equations in Three Variables Using DeterminantsCramer’s RuleIf

then

x =

Dx

D, y =

Dy

D, and z =

Dz

D, where D Z 0.

c a1x + b1y + c1z = d1

a2x + b2y + c2z = d2

a3x + b3y + c3z = d3

Evaluate the second-order determinant whose numerical factor is not 0.

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Section 8.5 Determinants and Cramer’s Rule 863

Using Cramer’s Rule to Solve a Linear System in Three Variables

Use Cramer’s rule to solve:

Solution Because

we need to set up and evaluate four determinants.

Step 1 Set up the determinants.

1. the determinant in all three denominators, consists of the and

2. the determinant in the numerator for is obtained by replacing thein 1, 1, and 2, with the constants on the right sides of the

equations, and 3.

3. the determinant in the numerator for is obtained by replacing thein 2, 4, and 3, with the constants on the right sides of the

equations, and 3.

4. the determinant in the numerator for is obtained by replacing thein and 1, with the constants on the right sides of the

equations, and 3.

Dz = 3 1 2 -41 4 -62 3 3

3-4, -6,

D, -1, -2,z-coefficientsz,Dz ,

Dy = 3 1 -4 -11 -6 -22 3 1

3-4, -6,

D,y-coefficientsy,Dy ,

Dx = 3 -4 2 -1-6 4 -2

3 3 1

3-4, -6,

D,x-coefficientsx,Dx ,

D = 3 1 2 -11 4 -22 3 1

3z-coefficients.

y-,x-,D,

x =

Dx

D, y =

Dy

D, and z =

Dz

D,

c x + 2y - z = -4x + 4y - 2z = -6

2x + 3y + z = 3.

EXAMPLE 5

Replace in with the constants on the rightof the three equations.

Dx-coefficients

Dy = 3 a1 d1 c1

a2 d2 c2

a3 d3 c3

3 Replace in with the constants on the rightof the three equations.

Dy-coefficients

Dz = 3 a1 b1 d1

a2 b2 d2

a3 b3 d3

3 Replace in with the constants on the rightof the three equations.

Dz-coefficients

These four third-order determinants are given by

These are the coefficients of the variables , and

Dx = 3 d1 b1 c1

d2 b2 c2

d3 b3 c3

3z.x, y D = 3 a1 b1 c1

a2 b2 c2

a3 b3 c3

3

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 863

DiscoveryWrite a system of two equations that is inconsistent. Now use determinants and the resultboxed above to verify that this is truly an inconsistent system. Repeat the same process for asystem with two dependent equations.

864 Chapter 8 Matrices and Determinants

Step 2 Evaluate the four determinants.

Using the same technique to evaluate each determinant, we obtain

Step 3 Substitute these four values and solve the system.

The solution can be checked by substitution into the original threeequations. The solution set is

Check Point 5 Use Cramer’s rule to solve the system:

Cramer’s Rule with Inconsistent and Dependent SystemsIf the determinant in the denominator, is 0, the variables described by thequotient of determinants are not real numbers. However, when thisindicates that the system is either inconsistent or contains dependent equations.Thisgives rise to the following two situations:

D = 0,D,

c 3x - 2y + z = 162x + 3y - z = -9

x + 4y + 3z = 2.

51-2, 1, 426.1-2, 1, 42

z =

Dz

D=

205

= 4

y =

Dy

D=

55

= 1

x =

Dx

D=

-105

= -2

Dx = -10, Dy = 5, and Dz = 20.

= 11102 - 1152 + 2102 = 5

= 114 + 62 - 112 + 32 + 21-4 + 42

=1D= -133 2 24

3

1

1

2

2

4

3

–1

–2

1

–2

12 22

3–1

1± 2 2 22

4–1

–2

Study TipTo find and you’ll needto apply the evaluation process for a

determinant three times. Thevalues of and cannot beobtained from the numbers thatoccur in the computation of D.

DzDx , Dy ,3 * 3

Dz ,Dx , Dy ,

Determinants: Inconsistent and Dependent Systems

1. If and at least one of the determinants in the numerator is not 0, thenthe system is inconsistent. The solution set is

2. If and all the determinants in the numerators are 0, then the equationsin the system are dependent. The system has infinitely many solutions.

D = 0

�.D = 0

� Use determinants to identifyinconsistent systems andsystems with dependentequations.

Although we have focused on applying determinants to solve linear systems,they have other applications, some of which we consider in the exercise set thatfollows Example 6.

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Section 8.5 Determinants and Cramer’s Rule 865

The Determinant of Any MatrixThe determinant of a matrix with rows and columns is said to be an determinant. The value of an determinant can be found in termsof determinants of order For example, we found the value of a third-orderdeterminant in terms of determinants of order 2.

We can generalize this idea for fourth-order determinants and higher. Wehave seen that the minor of the element is the determinant obtained by deletingthe row and the column in the given array of numbers. The cofactor of theelement is times the minor of . If the sum of the row andcolumn is even, the cofactor is the same as the minor. If the sum of the row andcolumn is odd, the cofactor is the opposite of the minor.

Let’s see what this means in the case of a fourth-order determinant.

Evaluating the Determinant of a Matrix

Evaluate the determinant of the following matrix:

A = D1 -2 3 0

-1 1 0 20 2 0 -32 3 -4 1

T .

4 : 4EXAMPLE 6

1i + j21i + j2

aij1-12i + jaij

jthithaij

n - 1.1n 7 22nth-order

nth-ordernn

n : n� Evaluate higher-orderdeterminants.

With two 0s in the third column, wewill expand along the third column.

Solution

ƒ A ƒ = 41 -2 3 0

-1 1 0 20 2 0 -32 3 -4 1

4

The determinant that follows 3 isobtained by crossing out the row andthe column (row 1, column 3) in theoriginal determinant. The minor for is obtained in a similar manner.

-4

= 3 3 -1 1 20 2 -32 3 1

3 + 4 3 1 -2 0-1 1 2

0 2 -3

33 is in row 1,

column 3.−4 is in row 4,

column 3.

–1

0

2

1

2

3

2

–3

1=(–1)1 ± 3(3) +(–1)4 ± 3(–4)3 3 1

–1

0

–2

1

2

0

2

–33 3

Evaluate the two third-order determinants to get

Check Point 6 Evaluate the determinant of the following matrix:

If a linear system has equations, Cramer’s rule requires you to computedeterminants of order. The excessive number of calculations required

to perform Cramer’s rule for systems with four or more equations makes it aninefficient method for solving large systems.

nthn + 1n

A = D0 4 0 -3

-1 1 5 21 -2 0 63 0 0 1

T .

ƒ A ƒ = 31-252 + 41-12 = -79.

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 865

866 Chapter 8 Matrices and Determinants

Exercise Set 8.5

Practice ExercisesEvaluate each determinant in Exercises 1–10.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

For Exercises 11–26, use Cramer’s rule to solve each system orto determine that the system is inconsistent or containsdependent equations.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

Evaluate each determinant in Exercises 27–32.

27. 28.

29. 30.

31. 32.

In Exercises 33–40, use Cramer’s rule to solve each system.

33. 34. c x - y + 2z = 32x + 3y + z = 9-x - y + 3z = 11

c x + y + z = 02x - y + z = -1-x + 3y - z = -8

3 1 2 32 2 -33 2 1

33 1 1 12 2 2

-3 4 -5

33 2 -4 2-1 0 5

3 0 4

33 3 1 0-3 4 0-1 3 -5

33 4 0 03 -1 42 -3 5

33 3 0 02 1 -52 5 -1

3

b2x = 7 + 3y

4x - 6y = 3b4y = 16 - 3x

6x = 32 - 8y

bx + 2y - 3 = 012 = 8y + 4x

b3x = 2 - 3y

2y = 3 - 2x

b y = -4x + 22x = 3y + 8

b2x = 3y + 25x = 51 - 4y

b3x = 7y + 12x = 3y - 1

b3x - 4y = 42x + 2y = 12

b2x - 9y = 53x - 3y = 11

b x + 2y = 35x + 10y = 15

b3x + 2y = 22x + 2y = 3

b4x - 5y = 172x + 3y = 3

b x - 2y = 55x - y = -2

b12x + 3y = 152x - 3y = 13

b2x + y = 3x - y = 3

bx + y = 7x - y = 3

`23

13

- 12

34``

12

12

18 -

34`

`15

16

-6 5``

-5 -1-2 -7

`

`1 -3

-8 2``

-7 142 -4

`

`7 9

-2 -5``

-4 15 6

`

`4 85 6

``5 72 3

`

35. 36.

37. 38.

39. 40.

Evaluate each determinant in Exercises 41–44.

41. 42.

43. 44.

Practice PlusIn Exercises 45–46, evaluate each determinant.

45. 46.

In Exercises 47–48, write the system of linear equations for whichCramer’s rule yields the given determinants.

47.

48.

In Exercises 49–52, solve each equation for

49. 50.

51. 52.

Application ExercisesDeterminants are used to find the area of a triangle whose verticesare given by three points in a rectangular coordinate system. Thearea of a triangle with vertices and is

where the symbol indicates that the appropriate sign shouldbe chosen to yield a positive area. Use this information to workExercises 53–54.

;

Area = ; 12

3 x1 y1 1x2 y2 1x3 y3 1

3 ,1x3 , y321x1 , y12, 1x2 , y22,

3 2 x 1-3 1 0

2 1 4

3 = 393 1 x -23 1 10 -2 2

3 = -8

`x + 3 -6x - 2 -4

` = 28`-2 x

4 6` = 32

x.

D = `2 -35 6

` , Dx = `8 -3

11 6`

D = `2 -43 5

` , Dx = `8 -4

-10 5`

4 ` 5 04 -3

` `-1 0

0 -1`

`7 -54 6

` `4 1

-3 5`

44 ` 3 1-2 3

` `7 01 5

`

`3 00 7

` `9 -63 5

`

4

4 1 -3 2 0-3 -1 0 -2

2 1 3 12 0 -2 0

44 -2 -3 3 51 -4 0 01 2 2 -32 0 1 1

4

4 3 -1 1 2-2 0 0 0

2 -1 -2 31 4 2 3

44 4 2 8 -7-2 0 4 1

5 0 0 54 0 0 -1

4

c 3x + 2z = 45x - y = -44y + 3z = 22

c x + 2z = 42y - z = 5

2x + 3y = 13

c 2x + 2y + 3z = 104x - y + z = -55x - 2y + 6z = 1

c x + y + z = 4x - 2y + z = 7x + 3y + 2z = 4

c x - 3y + z = -2x + 2y = 8

2x - y = 1c 4x - 5y - 6z = -1

x - 2y - 5z = -122x - y = 7

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Section 8.5 Determinants and Cramer’s Rule 867

53. Use determinants to find the area of the triangle whose vertices are (2, 6), and

54. Use determinants to find the area of the triangle whose vertices are (1, 1), and

Determinants are used to show that three points lie on the sameline (are collinear). If

then the points and are collinear. If thedeterminant does not equal 0, then the points are not collinear. Usethis information to work Exercises 55–56.

55. Are the points and (12, 5) collinear?

56. Are the points (1, 0), and (11, 12) collinear?

Determinants are used to write an equation of a line passingthrough two points. An equation of the line passing through thedistinct points and is given by

Use this information to work Exercises 57–58.

57. Use the determinant to write an equation of the line passingthrough and Then expand the determinant,expressing the line’s equation in slope-intercept form.

58. Use the determinant to write an equation of the line passingthrough and (2, 4). Then expand the determinant,expressing the line’s equation in slope-intercept form.

Writing in Mathematics59. Explain how to evaluate a second-order determinant.

60. Describe the determinants and in terms of the coeffi-cients and constants in a system of two equations in twovariables.

61. Explain how to evaluate a third-order determinant.

62. When expanding a determinant by minors, when is itnecessary to supply minus signs?

63. Without going into too much detail, describe how to solve alinear system in three variables using Cramer’s rule.

64. In applying Cramer’s rule, what does it mean if

65. The process of solving a linear system in three variablesusing Cramer’s rule can involve tedious computation. Isthere a way of speeding up this process, perhaps usingCramer’s rule to find the value for only one of the vari-ables? Describe how this process might work, presenting aspecific example with your description. Remember thatyour goal is still to find the value for each variable in thesystem.

66. If you could use only one method to solve linear systems inthree variables, which method would you select? Explainwhy this is so.

Technology Exercises67. Use the feature of your graphing utility that evaluates the

determinant of a square matrix to verify any five of thedeterminants that you evaluated by hand in Exercises 1–10,27–32, or 41–44.

D = 0?

DyDx

1-1, 32

1-2, 62.13, -52

3 x y 1x1 y1 1x2 y2 1

3 = 0.

1x2 , y221x1 , y12

1-4, -62,

13, -12, 10, -32,

1x3 , y321x1 , y12, 1x2 , y22,

3 x1 y1 1x2 y2 1x3 y3 1

3 = 0,

111, -32.1-2, -32,

1-3, 52.13, -52,In Exercises 68–69, use a graphing utility to evaluate thedeterminant for the given matrix.

68. 69.

70. What is the fastest method for solving a linear system withyour graphing utility?

Critical Thinking ExercisesMake Sense? In Exercises 71–74, determine whether eachstatement makes sense or does not make sense, and explainyour reasoning.

71. I’m solving a linear system using a determinant that containstwo rows and three columns.

72. I can speed up the tedious computations required by Cramer’srule by using the value of to determine the value of

73. When using Cramer’s rule to solve a linear system, the num-ber of determinants that I set up and evaluate is the same asthe number of variables in the system.

74. Using Cramer’s rule to solve a linear system, I found thevalue of to be zero, so the system is inconsistent.

75. a. Evaluate:

b. Evaluate:

c. Evaluate:

d. Describe the pattern in the given determinants.

e. Describe the pattern in the evaluations.

76. Evaluate:

77. What happens to the value of a second-order determinant ifthe two columns are interchanged?

78. Consider the system

Use Cramer’s rule to prove that if the first equation of thesystem is replaced by the sum of the two equations, theresulting system has the same solution as the originalsystem.

79. Show that the equation of a line through andis given by the determinant equation in Exercises

57–58.1x2 , y22

1x1 , y12

a2x + b2y = c2 .

a1x + b1y = c1

52 0 0 0 00 3 0 0 00 0 2 0 00 0 0 1 00 0 0 0 4

5 .

4 a a a a

0 a a a

0 0 a a

0 0 0 a

4 .

3 a a a

0 a a

0 0 a

3 .`a a

0 a` .

D

Dx .D

E8 2 6 -1 02 0 -3 4 72 1 -3 6 -5

-1 2 1 5 -14 5 -2 3 -8

UD3 -2 -1 4

-5 1 2 72 4 5 0

-1 3 -6 5

T

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868 Chapter 8 Matrices and Determinants

Summary, Review, and TestSummary

Group Exercise80. We have seen that determinants can be used to solve linear

equations, give areas of triangles in rectangular coordinates, anddetermine equations of lines. Not impressed with theseapplications? Members of the group should research anapplication of determinants that they find intriguing. The groupshould then present a seminar to the class about this application.

Preview ExercisesExercises 81–83 will help you prepare for the material covered inthe first section of the next chapter.

81. Consider the equation

a. Set and find the

b. Set and find the

82. Divide both sides of by 400 and simplify.

83. Complete the square and write the circle’s equation instandard form:

Then give the center and radius of the circle and graph theequation.

x2+ y2

- 2x + 4y = 4.

25x2+ 16y2

= 400

y-intercepts.x = 0

x-intercepts.y = 0

x2

9+

y2

4= 1.

DEFINITIONS AND CONCEPTS EXAMPLES

8.1 Matrix Solutions to Linear Systems

a. Matrix row operations are described in the box on page 807. Ex. 1, p. 807

b. To solve a linear system using Gaussian elimination, begin with the system’s augmented matrix. Use matrixrow operations to get 1s down the main diagonal from upper left to lower right, and 0s below the 1s. Such amatrix is in row-echelon form. Details are in the box on page 808.

Ex. 2, p. 809;Ex. 3, p. 811

c. To solve a linear system using Gauss-Jordan elimination, use the procedure of Gaussian elimination, butobtain 0s above and below the 1s in the main diagonal from upper left to lower right. Such a matrix is inreduced row-echelon form. Details are in the box on page 813.

Ex. 4, p. 814

8.2 Inconsistent and Dependent Systems and Their Applications

a. If Gaussian elimination results in a matrix with a row containing all 0s to the left of the vertical line and anonzero number to the right, the system has no solution (is inconsistent).

Ex. 1, p. 818

b. In a square system, if Gaussian elimination results in a matrix with a row with all 0s, but not a row like the onein part (a), the system has an infinite number of solutions (contains dependent equations).

Ex. 2, p. 820

c. In nonsquare systems, the number of variables differs from the number of equations. Ex. 3, p. 821

8.3 Matrix Operations and Their Applications

a. A matrix of order has rows and columns. Two matrices are equal if and only if they have the sameorder and corresponding elements are equal.

nmm * n Ex. 1, p. 827

b. Matrix Addition and Subtraction: Matrices of the same order are added or subtracted by adding or subtractingcorresponding elements. Properties of matrix addition are given in the box on page 830.

Ex. 2, p. 829

c. Scalar Multiplication: If is a matrix and is a scalar, then is the matrix formed by multiplying each elementin by Properties of scalar multiplication are given in the box on page 831.c.A

cAcA Ex. 3, p. 830;Ex. 4, p. 831

d. Matrix Multiplication: The product of an matrix and an matrix is an matrix Theelement in the row and column of is found by multiplying each element in the row of by thecorresponding element in the column of and adding the products. Matrix multiplication is not commutative:

Properties of matrix multiplication are given in the box on page 836.AB Z BA.Bjth

AithABjthithAB.m * pBn * pAm * n Ex. 5, p. 832;

Ex. 6, p. 833;Ex. 7, p. 835

8.4 Multiplicative Inverses of Matrices and Matrix Equations

a. The multiplicative identity matrix is an matrix with 1s down the main diagonal from upper left tolower right and 0s elsewhere.

n * nIn

b. Let be an square matrix. If there is a square matrix such that and thenis the multiplicative inverse of A.A-1

A-1 A = In ,AA-1

= InA-1n * nA Ex. 1, p. 843

c. If a square matrix has a multiplicative inverse, it is invertible or nonsingular. Methods for finding multiplicativeinverses for invertible matrices, including a formula for matrices, are given in the box on page 849.2 * 2

Ex. 2, p. 844;Ex. 3, p. 846;Ex. 4, p. 848

8C h a p t e r

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