PowerPoint for 7.5

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7.5 Determinants and Cramer’s Rule
• Subscript notation for the matrix A
 a11
a
 21
A   a31
 

am1
a12
a13

a22
a23

a32
a33



am 2
am 3

a1n 
a2 n 

a3n 
 

amn 
The row 1, column 1 element is a11; the row 2,
column 3 element is a23; and, in general, the row i,
column j element is aij.
Copyright © 2011 Pearson Education, Inc.
Slide 7.5-1
7.5 Determinants of 2 × 2 Matrices
• Associated with every square matrix is a real
number called the determinant of A. In this text,
we use det A.
The determinant of a 2 × 2 matrix A, where
a11 a12 

A
a21 a22 
is a real number defined as
det A  a11a22  a21a12 .
Copyright © 2011 Pearson Education, Inc.
Slide 7.5-2
7.5 Determinants of 2 × 2 Matrices
Example
Find det A if A   3 4.
 6 8
Analytic Solution
Graphing Calculator Solution
det A  det  3 4
 6 8
 3(8)  6(4)
 48
Copyright © 2011 Pearson Education, Inc.
Slide 7.5-3
7.5 Determinant of a 3 × 3 Matrix
The determinant of a 3 × 3 matrix A, where
 a11 a12 a13 
A  a21 a22 a23 
a a a 
 31 32 33 
is a real number defined as
det A  a11a22 a33  a12 a23a31  a13a21a32
  a31a22 a13  a32 a23a11  a33a21a12  .
Copyright © 2011 Pearson Education, Inc.
Slide 7.5-4
7.5 Determinant of a 3 × 3 Matrix
• A method for calculating 3 × 3 determinants is
found by re-arranging and factoring this formula.
 a11 a12 a13 
det A  a21 a22 a23 
a a a 
 31 32 33 
 a11 (a22 a33  a32 a23 )  a21 (a12 a33  a32 a13 )
 a31 (a12 a23  a22 a13 )
Each of the quantities in parentheses represents the
determinant of a 2 × 2 matrix that is part of the
3 × 3 matrix remaining when the row and column
of the multiplier are eliminated.
Copyright © 2011 Pearson Education, Inc.
Slide 7.5-5
7.5 The Minor of an Element
• The determinant of each 3 × 3 matrix is called a
minor of the associated element.
• The symbol Mij represents the minor when the ith
row and jth column are eliminated.
Copyright © 2011 Pearson Education, Inc.
Slide 7.5-6
7.5 The Cofactor of an Element
Let Mij be the minor for element aij in an n × n matrix. The
cofactor of aij, written Aij, is
Aij   1
i j
•
 M ij .
To find the determinant of a 3 × 3 or larger square matrix:
1. Choose any row or column,
2. Multiply the minor of each element in that row or
column by a +1 or –1, depending on whether the sum
of i + j is even or odd,
3. Then, multiply each cofactor by its corresponding
element in the matrix and find the sum of these
products. This sum is the determinant of the matrix.
Copyright © 2011 Pearson Education, Inc.
Slide 7.5-7
7.5 Finding the Determinant
 2  3  2
Example Evaluate det  1  4  3 , expanding
0
2
 1
by the second column.
Solution First find the minors of each element in the
second column.
M 12  det  1
 1
M 22  det  2
 1
M 32  det  2
 1
Copyright © 2011 Pearson Education, Inc.
 3  1(2)  (1)( 3)  5
2
 2  2(2)  (1)( 2)  2
2
 2  2(3)  (1)( 2)  8
 3
Slide 7.5-8
7.5 Finding the Determinant
Now, find the cofactor.
A12  (1)1 2  M 12  (1)3  (5)  5
A22  (1) 2 2  M 22  (1) 4  (2)  2
A32  (1)3 2  M 32  (1)5  (8)  8
The determinant is found by multiplying each cofactor by its
corresponding element in the matrix and finding the sum of
these products.
 2  3  2
det  1  4  3  a12  A12  a22  A22  a32  A32
2
 1 0
 3(5)  (4)(2)  (0)(8)
 23
Copyright © 2011 Pearson Education, Inc.
Slide 7.5-9
7.5 Cramer’s Rule for 2 × 2 Systems
The solution of the system
a1 x  b1 y  c1
a2 x  b2 y  c2 ,
Dx
is given by x 
D
where
and
Dy
y ,
D
 a1
 c1 b1 
Dx  det 
, Dy  det 

 a2
c2 b2 
Copyright © 2011 Pearson Education, Inc.
c1 
,

c2 
 a1 b1 
and D  det 
 0.

 a2 b2 
Slide 7.5-10
7.5 Applying Cramer’s Rule to a System
with Two Equations
Example Use Cramer’s rule to solve the system.
5x  7 y   1
6x  8 y  1
Analytic Solution By Cramer’s rule, x 
Dx
D
and y 
Dy
D
.
D  det 5 7   5(8)  6(7)  2
6 8 

1
7

  (1)(8)  (1)(7)  15
Dx  det
 1 8
5

1

  (5)(1)  (6)( 1)  11
Dy  det
6 1
Copyright © 2011 Pearson Education, Inc.
Slide 7.5-11
7.5 Applying Cramer’s Rule to a System
with Two Equations
Dx  15 15
x


and
D 2 2
Dy 11
11
y


D 2
2
The solution set is 152 , 112 .
Graphing Calculator Solution
Enter D, Dx, and Dy as matrices A, B, and C,
respectively.
Copyright © 2011 Pearson Education, Inc.
Slide 7.5-12
7.5 Cramer’s Rule for 3 × 3 Systems
The solution for the system
a1 x  b1 y  c1 z  d1
a2 x  b2 y  c2 z  d 2
a3 x  b3 y  c3 z  d3 ,
is given by x 
Dx
,
D
y
Dy
D
, and z  z ,
D
D
where
 d1
Dx  det  d 2
 d3
 a1
Dz  det  a2
 a3
Copyright © 2011 Pearson Education, Inc.
b1
b2
b3
b1
b2
b3
c1 
c2  ,
c3 
 a1 d1 c1 
Dy  det  a2 d 2 c2  ,
 a3 d3 c3 
d1 
 a1 b1 c1 
d 2  , and D  det  a2 b2 c2   0.
 a3 b3 c3 
d3 
Slide 7.5-13
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