4.6
4.6
In this
section
●
●
●
Minors
(4-39)
237
CRAMER’S RULE FOR SYSTEMS IN
THREE VARIABLES
The solution of linear systems involving three variables using determinants is very
similar to the solution of linear systems in two variables using determinants.
However, you first must learn to find the determinant of a 3 3 matrix.
Minors
Evaluating a 3 3
Determinant
To each element of a 3 3 matrix there corresponds a 2 2 matrix that is
obtained by deleting the row and column of that element. The determinant of
the 2 2 matrix is called the minor of that element.
Cramer’s Rule
E X A M P L E
Cramer’s Rule for Systems in Three Variables
1
Finding minors
Find the minors for the elements 2, 3, and 6 of the 3 3 matrix
1
2
6
2
0
4
8
3 .
7
Solution
To find the minor for 2, delete the first row and first column of the matrix:
Now find the determinant of
2
1
2
6
2
2
0
4
8
3
7
6 7:
3
6 7 (2)(7) (6)(3) 4
3
The minor for 2 is 4. To find the minor for 3, delete the second row and third column
of the matrix:
2 1 8
0 2
3
4 6
7
Now find the determinant of
4
2
4
2
1
:
6
1
(2)(6) (4)(1) 8
6
The minor for 3 is 8. To find the minor for 6, delete the third row and the second
column of the matrix:
2
0
4
1
2
6
8
3
7
238
(4-40)
Chapter 4
Systems of Linear Equations
Now find the determinant of
8
:
3
0
2
8
(2)(3) (0)(8) 6
3
0
2
The minor for 6 is 6.
■
Evaluating a 3 3 Determinant
study
The determinant of a 3 3 matrix is defined in terms of the determinants of
minors.
tip
Remember that everything
we do in solving problems is
based on principles (which are
also called rules, theorems,
and definitions). These principles justify the steps we take.
Be sure that you understand
the reasons. If you just memorize procedures without understanding, you will soon forget the procedures.
E X A M P L E
2
Determinant of a 3 3 Matrix
The determinant of a 3 3 matrix is defined as follows:
a1
a2
a3
b1
b2
b3
c1
b c
b
c2 a1 2 2 a2 1
b3 c3
b3
c3
c1
b c
a3 1 1
c3
b2 c2
Note that the determinants following a1, a2, and a3 are the minors for a1, a2,
and a3, respectively. Writing the determinant of a 3 3 matrix in terms of minors
is called expansion by minors. In the definition we expanded by minors about the
first column. Later we will see how to expand by minors using any row or column
and get the same value for the determinant.
Determinant of a 3 3 matrix
Find the determinant of the matrix by expansion by minors about the first column.
1
2
0
3
4
7
5
6
9
Solution
1
3 5
3 5
3 5
4 6
2
4
6 1
(2) 0
7
9
4
6
7 9
0 7
9
1 [36 (42)] 2 (27 35) 0 [18 (20)]
1 78 2 (8) 0
78 16
■
62
In the next example we evaluate a determinant using expansion by minors
about the second row. In expanding about any row or column, the signs of the
coefficients of the minors alternate according to the sign array that follows:
The sign array is easily remembered by observing that there is a “” sign in the
upper left position and then alternating signs for all of the remaining positions.
4.6
E X A M P L E
3
A calculator is very useful for
finding the determinant of a
3 3 matrix. Define A using
MATRX EDIT.
239
Determinant of a 3 3 matrix
Evaluate the determinant of the matrix by expanding by minors about the second
row.
calculator
close-up
(4-41)
Cramer’s Rule for Systems in Three Variables
1
3 5
2
4
6
0 7
9
Solution
For expansion using the second row we prefix the signs “ ” from the second
row of the sign array to the corresponding numbers in the second row of the matrix,
2, 4, and 6. Note that the signs from the sign array are used in addition to any
signs that occur on the numbers in the second row.
From the sign array, second row
Now use the determinant
function from MATRX MATH
and the A from MATRX NAMES
to find the determinant.
1 3 5
3
2 4
6 (2) 7
0 7
9
5
1
4
9
0
5
1
6
9
0
3
7
2(27 35) 4(9 0) 6(7 0)
2(8) 4(9) 6(7)
16 36 42
62
Note that 62 is the same value that was obtained for this determinant in
■
Example 2.
It can be shown that expanding by minors using any row or column prefixed by
the corresponding signs from the sign array yields the same value for the determinant. Because we can use any row or column to evaluate a determinant of a 3 3
matrix, we can choose a row or column that makes the work easier. We can shorten
the work considerably by picking a row or column with zeros in it.
E X A M P L E
4
Choosing the simplest row or column
Find the determinant of the matrix
3
4
7
5
6
9
0
0
2
Solution
We choose to expand by minors about the third column of the matrix because the
third column contains two zeros. Prefix the third-column entries 0, 0, 2 by the signs
“ ” from the third column of the sign array:
3
4
7
5 0
4 6
3 5
3
0
2
6 0 0 7
9
7
9
4
9 2
0 0 2[18 (20)]
4
5
6
■
240
(4-42)
Chapter 4
Systems of Linear Equations
Cramer’s Rule
A system of three linear equations in three variables can be solved by using determinants in a manner similar to that of the previous section. This rule is also called
Cramer’s rule.
Cramer’s Rule for Three Equations in Three Unknowns
The solution to the system
a1x b1y c1z d1
a2 x b2 y c2z d2
a3 x b3 y c3 z d3
Dy
D
is given by x Dx, y D, and z Dz
,
D
where
a1 b1
D a2 b2
a3 b3
c1
c2 ,
c3
d1
D x d2
d3
c1
c2 ,
c3
a1
D z a2
a3
a1 d1
D y a2 d2
a3 d3
b1
b2
b3
b1 d1
b2 d2 ,
b3 d3
c1
c2 ,
c3
provided that D 0.
Note that Dx , D y , and D z are obtained from D by replacing the x-, y-, or
z-column with the constants d1, d 2, and d 3.
E X A M P L E
5
Solving an independent system
Use Cramer’s rule to solve the system:
x yz4
x y
3
x 2y z 0
Solution
We first calculate D, D x , D y , and D z . To calculate D, expand by minors about the
third column because the third column has a zero in it:
1
D 1
1
1
1
1
1
0 1
1
2 1
1
1
0
2
1
1
1
(1) 2
1
1 [2 (1)] 0 (1)[1 1]
302
5
1
1
4.6
(4-43)
Cramer’s Rule for Systems in Three Variables
241
For Dx , expand by minors about the first column:
calculator
4
Dx 3
0
close-up
When you see the amount of
arithmetic required to solve
the system in Example 5 by
Cramer’s rule, you can understand why computers and
calculators have been programmed to perform this
method. Some calculators can
find determinants for matrices as large as 10 10. Try
to solve Example 5 with a
graphing calculator that has
determinants.
1
1
2
1
1
0 4
2
1
0
1
(3) 1
2
1
1
0
1
1
1
0
4 (1 0) 3 (1 2) 0
4 9 0 5
For Dy , expand by minors about the third row:
1
Dy 1
1
4
3
0
1
4
0 1
3
1
0 (1) 1
1
1
0
0
1
1
1
4
3
1
1
1 3 0 (1)(7) 10
To get Dz , expand by minors about the third row:
1
Dz 1
1
1
4
1
1 3 1 1
2
0
4
1
2
3
1
4
1
0
3
1
1 1 2(7) 0 15
Now, by Cramer’s rule,
Dx 5
x 1,
5
D
Dy 10
y 2,
D
5
Dz 15
z 3.
D
5
and
Check (1, 2, 3) in the original equations. The solution set is (1, 2, 3)
.
■
If D 0, Cramer’s rule does not apply. Cramer’s rule provides the solution
only to a system of three equations with three variables that has a single point in the
solution set. If D 0, then the solution set either is empty or consists of infinitely
many points, and we use elimination of variables to find the solution.
E X A M P L E
6
Solving a dependent system
Solve the system:
x y z2
2x 2y 2z 4
3x 3y 3z 6
(1)
(2)
(3)
Solution
Calculate D by expanding about the first column:
1
2
D
3
1
2
3
1
2 1 2
3
3
2
1
2
3
3
1
1
(3) 3
2
1
2
1 0 2 0 (3) 0 0
Because D 0, Cramer’s rule does not apply to this system. If we multiply Eq. (1)
by 2, we get Eq. (2). If we multiply Eq. (1) by 3, we get Eq. (3). Thus all three
equations are equivalent, and they are dependent. The solution set to the system is
■
(x, y, z)x y z 2
.
242
(4-44)
Chapter 4
WARM-UPS
Systems of Linear Equations
True or false? Explain your answer.
1. A minor is the determinant of a 2 2 matrix. True
2. The minor for an element is found by deleting the element from the matrix.
False
3. The determinant of a 3 3 matrix is found by using minors. True
4. Expansion by minors converts a 3 3 matrix into a 4 4 matrix. False
5. Using Cramer’s rule, we use D to get the value of x. False
Dx
6. Expansion by minors about any row or any column gives the same value for
the determinant of a 3 3 matrix. True
7. The sign array is used in evaluating the determinant of a 3 3 matrix.
True
8. It is easier to find the determinant of a 3 3 matrix with several zero
elements than one with no zero elements. True
9. If D 0, then x, y, and z are all zero. False
10. Cramer’s rule solves nonlinear systems of three equations in three
unknowns. False
4. 6
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What is a minor?
A minor for an element in a 3 3 matrix is the determinant of a 2 2 matrix.
9. Minor for 7 3
10. Minor for 0 1
11. Minor for 1 1
12. Minor for 6 1
Find the determinant of each 3 3 matrix by using expansion
by minors about the first column. See Example 2.
1
3
1
2
1
5
15.
1
0
1
0
1
2
4. Which systems can be solved by Cramer’s rule for three
equations in three unknowns?
Cramer’s rule solves only those systems that have a unique
solution.
17.
Find the indicated minors using the following matrix. See
Example 1.
19.
2. How do you find the minor for an element of a 3 3
matrix?
A minor for an element is obtained by deleting the row and
column of the element and finding the determinant of the
2 2 matrix that remains.
13.
3. What is the purpose of the sign array?
The sign array tells what signs to use in the expansion by
minors.
5. Minor for 3
7. Minor for 5
11
4
3
4
0
2
5
3
7
1 6
6. Minor for 2 24
8. Minor for 3 18
1
2
3
2
1
3
2
3
5
1
0
0
1
3
4
1
3
2
7
14.
1
9
2
1
0
5
2
3
5
1
1
4
3
2
6
1
16.
0
1
3
2
3
0
5
18.
20.
2
1
3
1
2
4
2
1
2
1
0
0
1
4
1
0
1
0
3
2
1
6
4
9
26
9
Evaluate the determinant of each 3 3 matrix using expansion by minors about the row or column of your choice. See
Examples 3 and 4.
21.
3
2
4
1
0
0
5
6
1
22
22.
2
1
3
1
2
0
2
5
0
3
4.6
23.
2
1
3
0
1 1
2 4 3
2 3 0
4 1 0
0
3 5
25.
27.
2
0
5
1
0
0
1
5
4
6
70
25
2
0
3
2
4 2
26.
2
0
1
28.
24.
2
6
1
3
4
2
1
5
6
6 3
4 0
4 5
0
1
0
6
28
1
(4-45)
Cramer’s Rule for Systems in Three Variables
243
Write a system of three equations in three variables for each
word problem. Use Cramer’s rule to solve each system.
45. Weighing dogs. Cassandra wants to determine the weights
of her two dogs, Mimi and Mitzi. However, neither dog will
sit on the scale by herself. Cassandra, Mimi, and Mitzi
altogether weigh 175 pounds. Cassandra and Mimi together
weigh 143 pounds. Cassandra and Mitzi together weigh
139 pounds. How much does each weigh individually?
Mimi 36 pounds, Mitzi 32 pounds, Cassandra 107 pounds
175
143
139
Cassandra
Mimi
Mitzi
Cassandra
Mimi
Cassandra
Mitzi
Use Cramer’s rule to solve each system. See Example 5.
29.
xyz6
xyz2
2x y z 7
(1, 2, 3)
30.
xy z2
x y 2z 3
2x y z 7
(1, 2, 3)
31. x 3y 2z 0
x y z2
x y z0
(1, 1, 2)
32. 3x 2y 2z 0
x y z1
x y z3
(2, 1, 2)
33. x y
1
2y z 3
x yz0
(3, 2, 1)
34. x y
8
x
2z 0
xy z1
(6, 2, 3)
35.
x yz0
2x 2y z 6
x 3y
0
3 1
, , 2
2 2
36.
xy z0
2y 2z 0
3x y
1
(0, 1, 1)
38. x
z0
x 3y
1
4y 3z 3
(1, 0, 1)
37.
xy z1
5x y
0
3x y 2z 0
1 5
, , 2
2 2
Solve each system. Use Cramer’s rule if possible. See Example 6.
39.
2x y z 1
40. x y z 4
6x 3y 3z 3
2x 2y 2z 3
4x 2y 2z 2
4x y z 1
(x, y, z) 2x y z 1
41. x y
1
y 2z 3
x 2y 2z 5
42.
x y z5
2x 2y 2z 10
3x 3y 3z 15
(x, y, z) x y z 5
43. x y
4
y z 3
x
z 5
(1, 3, 6)
44. x y
0
x
z 1
yz3
FIGURE FOR EXERCISE 45
46. Nickels, dimes, and quarters. Bernard has 41 coins consisting of nickels, dimes, and quarters, and they are worth a
total of $4.00. If the number of dimes plus the number of
quarters is one more than the number of nickels, then how
many of each does he have?
20 nickels, 15 dimes, 6 quarters
47. Finding three angles. If the two acute angles of a right triangle differ by 12°, then what are the measures of the three
angles of this triangle?
39°, 51°, 90°
48. Two acute and one obtuse. The obtuse angle of a triangle
is twice as large as the sum of the two acute angles. If the
smallest angle is only one-eighth as large as the sum of the
other two, then what is the measure of each angle?
20°, 40°, 120°
GET TING MORE INVOLVED
49. Writing. For what values of a, b, c, and d is the determinant of the matrix
a b 0
c d 0
b a 0
equal to zero? Explain your answer.
50. Exploration. The determinant of a 4 4 matrix is found
by expanding by 3 3 minors and using a 4 4 sign
array. Find the determinant of a 4 4 matrix of your
choice. Use the definition of the determinant of a 3 3
matrix as a guide.
244
(4-46)
Chapter 4
Systems of Linear Equations
GR APHING C ALCUL ATOR
EXERCISES
51. Use the determinant feature of your graphing calculator to
find the determinant of each matrix in Exercises 13–20 of
this section.
52. Solve the systems in Exercises 29–38 by using Cramer’s
rule and your graphing calculator to find the determinants.
4.7
53. The solution to an independent system of four linear equations in four variables can be found by using determinants
of 4 4 matrices in the same manner as Cramer’s rule for
three variables. Write Cramer’s rule for four variables.
Make up a system of four linear equations in four variables
and solve it using your new rule and a graphing calculator
to evaluate the determinants.
LINEAR PROGRAMMING
●
Graphing the Constraints
In this section we graph the solution set to a system of several linear inequalities,
much as we graphed compound linear inequalities in Chapter 3. We then use the
solution set as the domain of a function for which we are seeking the maximum or
minimum value. The method that we use is called linear programming, and it
can be applied to problems such as finding maximum profit or minimum cost.
●
Maximizing or Minimizing a
Linear Function
Graphing the Constraints
In this
section
In linear programming we have two variables that must satisfy several linear
inequalities. These inequalities are called the constraints because they restrict the
variables to only certain values. A graph in the coordinate plane is used to indicate
the points that satisfy all of the constraints.
E X A M P L E
1
x 0, y 0
3x 2y 12
x 2y 8
y
5
(0, 4)
3
2
x 2y 8
(2, 3)
3x 2y 12
1
–1
–1
–2
1 2 3 (4, 0)
Graphing the constraints
Graph the solution set to the system of inequalities and identify each vertex of the
region:
x
FIGURE 4.7
Solution
The points on or to the right of the y-axis satisfy x 0. The points on or above the
x-axis satisfy y 0. The points on or below the line 3x 2y 12 satisfy
3x 2y 12. The points on or below the line x 2y 8 satisfy x 2y 8.
Graph each straight line and shade the region that satisfies all four inequalities as
shown in Fig. 4.7. Three of the vertices are easily identified as (0, 0), (0, 4),
and (4, 0). The fourth vertex is found by solving the system 3x 2y 12 and
■
x 2y 8. The fourth vertex is (2, 3).
In linear programming the constraints usually come from physical limitations in
some problem. In the next example we write the constraints and then graph the
points in the coordinate plane that satisfy all of the constraints.
E X A M P L E
2
Writing the constraints
Jules is in the business of constructing dog houses. A small dog house requires
8 square feet (ft2) of plywood and 6 ft2 of insulation. A large dog house requires
16 ft2 of plywood and 3 ft2 of insulation. Jules has available only 48 ft2 of plywood