Download 10.3 Systems of Linear Equations: Determinants

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CHAPTER 10
10.3
Systems of Equations and Inequalities
Systems of Linear Equations: Determinants
OBJECTIVES
1
2
3
4
5
Evaluate 2 by 2 Determinants
Use Cramer’s Rule to Solve a System of Two Equations Containing Two Variables
Evaluate 3 by 3 Determinants
Use Cramer’s Rule to Solve a System of Three Equations Containing Three
Variables
Know Properties of Determinants
In the preceding section, we described a method of using matrices to solve a system
of linear equations. This section deals with yet another method for solving systems
of linear equations; however, it can be used only when the number of equations
equals the number of variables. Although the method will work for any system (provided that the number of equations equals the number of variables), it is most often
used for systems of two equations containing two variables or three equations containing three variables. This method, called Cramer’s Rule, is based on the concept of
a determinant.
1 Evaluate 2 by 2 Determinants
✓
If a, b, c, and d are four real numbers, the symbol
D = `
a b
`
c d
is called a 2 by 2 determinant. Its value is the number ad - bc; that is,
D = `
a b
` = ad - bc
c d
(1)
The following device may be helpful for remembering the value of a 2 by 2
determinant:
bc
a
b
c
d
ad bc
Minus
EXAMPLE 1
ad
Evaluating a 2 : 2 Determinant
Evaluate:
`
3
6
-2
`
1
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SECTION 10.3
Algebraic Solution
3
`
6
759
Graphing Solution
First, we enter the matrix whose entries are those of the determinant into
the graphing utility and name it A. Using the determinant command, we
obtain the result shown in Figure 10.
-2
` = 132112 - 1621-22
1
= 3 - 1-122
= 15
Systems of Linear Equations: Determinants
Figure 10
NOW WORK PROBLEM
7.
2 Use Cramer’s Rule to Solve a System of Two Equations
✓
Containing Two Variables
Let’s now see the role that a 2 by 2 determinant plays in the solution of a system of
two equations containing two variables. Consider the system
ax + by = s
cx + dy = t
b
(1)
(2)
(2)
We shall use the method of elimination to solve this system.
Provided d Z 0 and b Z 0, this system is equivalent to the system
b
adx + bdy = sd
bcx + bdy = tb
(1) Multiply by d.
(2)
Multiply by b.
On subtracting the second equation from the first equation, we get
b
1ad - bc2x + 0 # y = sd - tb
bcx + bdy = tb
(1)
(2)
Now the first equation can be rewritten using determinant notation.
`
If D = `
a b
s b
`x = `
`
c d
t d
a b
` = ad - bc Z 0, we can solve for x to get
c d
x =
`
s b
`
t d
a b
`
`
c d
`
=
s b
`
t d
(3)
D
Return now to the original system (2). Provided that a Z 0 and c Z 0, the system is equivalent to
b
acx + bcy = cs
acx + ady = at
(1) Multiply by c.
(2) Multiply by a.
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CHAPTER 10
Systems of Equations and Inequalities
On subtracting the first equation from the second equation, we get
b
acx +
bcy
=
cs
0 # x + 1ad - bc2y = at - cs
(1)
(2)
The second equation can now be rewritten using determinant notation.
`
If D = `
a b
a s
`y = `
`
c d
c t
a b
` = ad - bc Z 0, we can solve for y to get
c d
`
y =
a s
`
c t
a b
`
`
c d
`
a s
`
c t
=
(4)
D
Equations (3) and (4) lead us to the following result, called Cramer’s Rule.
Theorem
Cramer’s Rule for Two Equations Containing Two Variables
The solution to the system of equations
b
ax + by = s
cx + dy = t
(1)
(2)
(5)
is given by
x =
`
s b
`
t d
a b
`
`
c d
,
y =
`
a s
`
c t
a b
`
`
c d
(6)
provided that
D = `
a b
` = ad - bc Z 0
c d
In the derivation given for Cramer’s Rule above, we assumed that none of the
numbers a, b, c, and d was 0. In Problem 60 you will be asked to complete the proof
under the less stringent condition that D = ad - bc Z 0.
Now look carefully at the pattern in Cramer’s Rule. The denominator in the
solution (6) is the determinant of the coefficients of the variables.
b
ax + by = s
cx + dy = t
D = `
a b
`
c d
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SECTION 10.3
Systems of Linear Equations: Determinants
761
In the solution for x, the numerator is the determinant, denoted by Dx , formed by
replacing the entries in the first column (the coefficients of x) of D by the constants
on the right side of the equal sign.
Dx = `
s b
`
t d
In the solution for y, the numerator is the determinant, denoted by Dy , formed by
replacing the entries in the second column (the coefficients in y) of D by the
constants on the right side of the equal sign.
Dy = `
a s
`
c t
Cramer’s Rule then states that, if D Z 0,
x =
EXAMPLE 2
Dx
,
D
y =
Dy
(7)
D
Solving a System of Linear Equations Using Determinants
Use Cramer’s Rule, if applicable, to solve the system
b
3x - 2y = 4
6x + y = 13
(1)
(2)
Algebraic Solution
Graphing Solution
The determinant D of the coefficients of the variables is
We enter the coefficient matrix into our graphing
utility. Call it A and evaluate det3A4. Since
det3A4 Z 0, we can use Cramer’s Rule. We enter the
matrices Dx and Dy into our graphing utility and call
them B and C, respectively. Finally, we find x
det3B4
det3C4
.
by calculating
and y by calculating
det3A4
det3A4
The results are shown in Figure 11.
D = `
3
6
-2
` = 132112 - 1621-22 = 15
1
Because D Z 0, Cramer’s Rule (7) can be used.
`
4
13
-2
`
1
Dx
=
D
15
142112 - 11321-22
=
15
30
=
15
= 2
x =
y =
Dy
`
3
6
4
`
13
=
D
15
1321132 - 162142
=
15
15
=
15
= 1
Figure 11
The solution is x = 2, y = 1.
In attempting to use Cramer’s Rule, if the determinant D of the coefficients of
the variables is found to equal 0 (so that Cramer’s Rule is not applicable), then the
system either is inconsistent or has infinitely many solutions.
NOW WORK PROBLEM
15.
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CHAPTER 10
Systems of Equations and Inequalities
3 Evaluate 3 by 3 Determinants
✓
To use Cramer’s Rule to solve a system of three equations containing three
variables, we need to define a 3 by 3 determinant.
A 3 by 3 determinant is symbolized by
a12
a22
a32
a11
3 a21
a31
a13
a23 3
a33
(8)
in which a11 , a12 , Á , are real numbers.
As with matrices, we use a double subscript to identify an entry by indicating its
row and column numbers. For example, the entry a23 is in row 2, column 3.
The value of a 3 by 3 determinant may be defined in terms of 2 by 2 determinants by the following formula:
Minus
a11
3 a21
a31
a12
a22
a32
a13
a
a23 3 = a11 ` 22
a32
a33
a23 ∂
a
` - a12 ` 21
a33
a31
q
2 by 2
determinant
left after
removing row
and column
containing a11
a 23
a
` + a13 ` 21
a 33
a31
q
2 by 2
determinant
left after
removing row
and column
containing a12
a22
`
a32
(9)
q
2 by 2
determinant
left after
removing row
and column
containing a13
The 2 by 2 determinants shown in formula (9) are called minors of the 3 by 3
determinant. For an n by n determinant, the minor Mij of entry aij is the determinant
resulting from removing the ith row and jth column.
EXAMPLE 3
Finding Minors of a 3 by 3 Determinant
2
3
For the determinant A = -2
0
Solution
-1
5
6
3
1 3 , find:
-9
(a) M12
(b) M23
(a) M12 is the determinant that results from removing the first row and second
column from A.
2
A = 3 -2
0
-1
5
6
3
13
-9
M12 = `
-2
0
1
` = 1-221-92 - 102112 = 18
-9
(b) M23 is the determinant that results from removing the second row and third column from A.
2
A = 3 -2
0
-1
5
6
3
13
-9
M23 = `
2
0
-1
` = 122162 - 1021-12 = 12
6
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SECTION 10.3
763
Systems of Linear Equations: Determinants
Referring back to formula (9), we see that each element aij is multiplied by its
minor, but sometimes this term is added and other times, subtracted. To determine
whether to add or subtract a term, we must consider the cofactor.
For an n by n determinant A, the cofactor of entry aij , denoted by A ij , is
given by
A ij = 1-12i + jMij
where Mij is the minor of entry aij .
The exponent of 1-12i + j is the sum of the row and column of the entry aij , so if
i + j is even, 1-12i + j will equal 1, and if i + j is odd, 1-12i + j will equal -1.
To find the value of a determinant, multiply each entry in any row or column by
its cofactor and sum the results. This process is referred to as expanding across a row
or column. For example, the value of the 3 by 3 determinant in formula (9) was
found by expanding across row 1.
If we choose to expand down column 2, we obtain
a11
3 a21
a31
a12
a22
a32
a13
a
a23 3 = 1-121 + 2 a12 ` 21
a31
a33
a23
a
` + 1-122 + 2a22 ` 11
a33
a31
a13
a
` + 1-123 + 2a32 ` 11
a33
a21
a13
`
a23
a13
a
` + 1-123 + 3a33 ` 11
a23
a21
a12
`
a22
æ
Expand down column 2.
If we choose to expand across row 3, we obtain
a11
3 a21
a31
a12
a22
a32
a13
a
a23 3 = 1-123 + 1 a31 ` 12
a22
a33
a13
a
` + 1-123 + 2a32 ` 11
a23
a21
æ
Expand across row 3.
It can be shown that the value of a determinant does not depend on the choice
of the row or column used in the expansion. However, expanding across a row or
column that has an element equal to 0 reduces the amount of work needed to compute the value of the determinant.
EXAMPLE 4
Evaluating a 3 : 3 Determinant
Find the value of the 3 by 3 determinant:
3
34
8
Solution
4
6
-2
-1
23
3
We choose to expand across row 1.
3
34
8
4
6
-2
-1
6
2 3 = 1-121 + 1 3 `
-2
3
2
4
` + 1-121 + 24 `
3
8
2
4
` + 1-121 + 31-12 `
3
8
= 3118 + 42 - 4112 - 162 + 1-121-8 - 482
= 31222 - 41-42 + 1-121-562
= 66 + 16 + 56 = 138
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`
-2
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CHAPTER 10
Systems of Equations and Inequalities
We could also find the value of the 3 by 3 determinant in Example 4 by
expanding down column 3.
3
34
8
4
6
-2
-1
4
2 3 = 1-121 + 31-12 `
8
3
6
3
` + 1-122 + 32 `
-2
8
4
3
` + 1-123 + 33 `
-2
4
4
`
6
= -11-8 - 482 - 21-6 - 322 + 3118 - 162
= 56 + 76 + 6 = 138
Evaluating 3 * 3 determinants on a graphing utility follows the same
procedure as evaluating 2 * 2 determinants.
NOW WORK PROBLEM
11.
4 Use Cramer’s Rule to Solve a System of Three Equations
✓
Containing Three Variables
Consider the following system of three equations containing three variables.
a11x + a12 y + a13 z = c1
c a21x + a22 y + a23 z = c2
a31x + a32 y + a33 z = c3
(10)
It can be shown that if the determinant D of the coefficients of the variables is not 0,
that is, if
a11 a12 a13
D = 3 a21 a22 a23 3 Z 0
a31 a32 a33
then the unique solution of system (10) is given by
Cramer’s Rule for Three Equations Containing Three Variables
x =
Dx
D
y =
Dy
D
z =
Dz
D
where
c1
Dx = 3 c2
c3
a12
a22
a32
a13
a23 3
a33
a11
Dy = 3 a21
a31
c1
c2
c3
a13
a23 3
a33
a11
Dz = 3 a21
a31
a12
a22
a32
c1
c2 3
c3
Do you see the similarity of this pattern and the pattern observed earlier for a
system of two equations containing two variables?
EXAMPLE 5
Using Cramer’s Rule
Use Cramer’s Rule, if applicable, to solve the following system:
2x + y - z = 3
c -x + 2y + 4z = -3
x - 2y - 3z = 4
(1)
(2)
(3)
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Solution
2
D = 3 -1
1
765
Systems of Linear Equations: Determinants
SECTION 10.3
The value of the determinant D of the coefficients of the variables is
1
2
-2
-1
2
4
-1
4 3 = 1-121 + 1 2 `
` + 1-121 + 21 `
-2 -3
1
-3
= 2122 - 11-12 + 1-12102
= 4 + 1 = 5
4
-1
` + 1-121 + 31-12 `
-3
1
2
`
-2
Because D Z 0, we proceed to find the values of Dx , Dy , and Dz .
3
Dx = 3 -3
4
1
2
-2
-1
2
4
-3
4 3 = 1-121 + 1 3 `
` + 1-121 + 21 `
-2 -3
4
-3
= 3122 - 11-72 + 1-121-22 = 15
4
-3
` + 1-121 + 31-12 `
-3
4
2
`
-2
2
Dy = 3 -1
1
3
-3
4
4
-1
` + 1-121 + 31-12 `
-3
1
-3
`
4
2
Dz = 3 -1
1
1
2
-2
-1
43 =
-3
=
=
3
-3 3 =
4
=
1-121 + 1 2 `
-3
4
4
-1
` + 1-121 + 23 `
-3
1
21-72 - 31-12 + 1-121-12
-14 + 3 + 1 = -10
1-121 + 1 2 `
2
-2
-3
-1
` + 1-121 + 21 `
4
1
-3
-1
` + 1-121 + 33 `
4
1
2
`
-2
2122 - 11-12 + 3102 = 5
As a result,
x =
Dx
15
=
= 3,
D
5
y =
Dy
D
=
-10
= -2,
5
z =
Dz
D
=
5
= 1
5
The solution is x = 3, y = -2, z = 1.
If the determinant of the coefficients of the variables of a system of three linear
equations containing three variables is 0, then Cramer’s Rule is not applicable. In
such a case, the system either is inconsistent or has infinitely many solutions.
Solving systems of three equations containing three variables using Cramer’s
Rule on a graphing utility follows the same procedure as that for solving systems of
two equations containing two variables.
NOW WORK PROBLEM
33.
5 Know Properties of Determinants
✓
Determinants have several properties that are sometimes helpful for obtaining their
value. We list some of them here.
Theorem
The value of a determinant changes sign if any two rows (or any two columns)
are interchanged.
(11)
Proof for 2 by 2 Determinants
`
a b
` = ad - bc and
c d
`
c d
` = bc - ad = -1ad - bc2
a b
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CHAPTER 10
Systems of Equations and Inequalities
EXAMPLE 6
Demonstrating Theorem (11)
`
Theorem
3
1
4
` =6-4=2
2
`
1
3
2
` = 4 - 6 = -2
4
If all the entries in any row (or any column) equal 0, the value of the
determinant is 0.
(12)
■
Proof Expand across the row (or down the column) containing the 0’s.
Theorem
If any two rows (or any two columns) of a determinant have corresponding
entries that are equal, the value of the determinant is 0.
(13)
You are asked to prove this result for a 3 by 3 determinant in which the entries
in column 1 equal the entries in column 3 in Problem 63.
EXAMPLE 7
Demonstrating Theorem (13)
1
31
4
Theorem
2
2
5
3
2 3
1 3
1
3 3 = 1-121 + 11 `
` + 1-121 + 22 `
` + 1-121 + 33 `
5 6
4 6
4
6
= 11-32 - 21-62 + 31-32 = -3 + 12 - 9 = 0
2
`
5
If any row (or any column) of a determinant is multiplied by a nonzero
number k, the value of the determinant is also changed by a factor of k. (14)
You are asked to prove this result for a 3 by 3 determinant using row 2 in
Problem 62.
EXAMPLE 8
Demonstrating Theorem (14)
`
`
Theorem
1
4
2
` = 6 - 8 = -2
6
k 2k
1
` = 6k - 8k = -2k = k1-22 = k `
4
6
4
2
`
6
If the entries of any row (or any column) of a determinant are multiplied by a
nonzero number k and the result is added to the corresponding entries of another
row (or column), the value of the determinant remains unchanged.
(15)
In Problem 64, you are asked to prove this result for a 3 by 3 determinant using
rows 1 and 2.
Copyright © 2006 Pearson Education, Inc., publishing as Pearson Prentice Hall
SECTION 10.3
EXAMPLE 9
Systems of Linear Equations: Determinants
767
Demonstrating Theorem (15)
`
3
5
4
` = -14
2
`
3
5
4
-7
`:`
2
5
0
` = -14
2
æ
Multiply row 2 by -2 and add to row 1.
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