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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
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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.
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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
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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.
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Determinants
The arrows in the following diagram will remind you which products to find when evaluating a 2 2 determinant.
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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
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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
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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.
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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.
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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
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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.
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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
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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.
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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
b. 3Solution
Here i = 2 and j = 3 so, 23
6 210.
1 2M
The cofactor is 2 3( 1) (10) 1(10) 10.
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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
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.
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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.
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Example 3 EVALUATING A 3 3 DETERMINANT
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.
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Example 3 EVALUATING A 3 3 DETERMINANT
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
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Example 3 EVALUATING A 3 3 DETERMINANT
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
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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.
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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,
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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
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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
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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.
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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
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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.
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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
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Example 4 APPLYING CRAMER’S RULE TO A 2 2 SYSTEM
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
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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
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Example 5 APPLYING CRAMER’S RULE TO A 3 3 SYSTEM
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.
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Example 5 APPLYING CRAMER’S RULE TO A 3 3 SYSTEM
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
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Example 5 APPLYING CRAMER’S RULE TO A 3 3 SYSTEM
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
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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.
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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.