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CHAPTER 6:
Systems of Equations
and Matrices
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
Systems of Equations in Two Variables
Systems of Equations in Three Variables
Matrices and Systems of Equations
Matrix Operations
Inverses of Matrices
Determinants and Cramer’s Rule
Systems of Inequalities and Linear Programming
Partial Fractions
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
6.4
Matrix Operations
Add, subtract, and multiply matrices when
possible.
Write a matrix equation equivalent to a system of
equations.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Matrices
A capital letter is generally used to name a matrix, and lowercase letters with double subscripts generally denote its
entries.
For example, a23 read “a sub two three,” indicates the entry in
the second row and the third column.
a11
A [aij ]
am1
a1n
amn
Two matrices are equal if they have the same order and
corresponding entries are equal.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 4
Matrix Addition and Subtraction
To add or subtract matrices, we add or subtract their
corresponding entries. The matrices must have the
same order.
Addition and Subtraction of Matrices
Given two m n matrices A = [aij] and B = [bij],
their sum is
A + B = [aij + bij]
and their difference is
A B = [aij bij].
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 5
Example
Find A + B for each of the following.
a)
6 7
A=
1
2
4
b)
1 4
A 2 6
7 0
2 4
B=
8 4
2 4
B 4 2
5 3
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 6
Example continued
We have a pair of 2 2 matrices in part (a) and a
pair of 3 2 matrices in part (b). Since each pair has
the same order we can add their corresponding
entries.
6 7 2 4
a) A + B = 1
2
8
4
4
6 2 7 (4)
1
2 (8) 4 (4)
4 3
3
6
3
4
b)
1 4 2 4
A + B 2 6 4 2
7 0 5 3
44
1 2
2 (4)
62
7 (5) 0 (3)
3 8
6 8
2 3
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 7
Examples
Find C D for each of the following.
a)
b)
1 2
1 3
C 3 0 D 4 7
4 2
0 3
5 6
C
1 2
9
D
2
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 8
Examples
a) Since the order of each matrix is 3 2, we can
subtract corresponding entries.
1 2 1 3
C D 3 0 4 7
4 2 0 3
1 (1) 2 (3) 2 5
3 4
0 7 7 7
4 0
2 3 4 5
b) Since the matrices do not have the same order, we
cannot subtract them.
5 6 D 9
C
2
1
2
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 9
Scalar Multiplication
When we find the product of a number and a
matrix, we obtain a scalar product.
The scalar product of a number k and a matrix A
is the matrix denoted kA, obtained by multiplying
each entry of A by the number k. The number k is
called a scalar.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 10
Example
4 1
Find 4A and (2)A for A =
.
0 7.
Solution:
4 1 4(4) 4(1) 16 4
4A = 4
0
7
4
(
0
)
4
(
7
)
0
28
4 1 2(4) 2(1) 8 2
2A = 2
0
7
2(0)
2(7)
0
14
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 11
Properties of Matrix Addition and Scalar
Multiplication
For any m n matrices, A, B, and C and any scalars
k and l:
Commutative Property of Addition
A + B = B + A.
Associative Property of Addition
A + (B + C) = (A + B) + C.
Associative Property of Scalar Multiplication
(kl)A = k(lA).
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 12
More Properties
Distributive Property
k(A + B) = kA + kB.
(k + l)A = kA + lA.
Additive Identity Property
There exists a unique matrix 0 such that:
A + 0 = 0 + A = A.
Additive Inverse Property
There exists a unique matrix A such that:
A + (A) = A + A = 0.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 13
Matrix Multiplication
For an m n matrix A = [aij] and an n p matrix
B = [bij], the product AB = [cij] is an m p matrix,
where
cij = ai1 • b1j + ai2 • b2j + ai3 • b3j + … + ain • bnj.
We can multiply two matrices only when the
number of columns in the first matrix is equal to the
number of rows in the second matrix.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 14
Examples
For
0 4
2
2
2
2 4
A
,
,
B 2 7 , and C
1
0
4
1 0
1 3
find each of the following.
a) AB
b) BA
c) AC
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 15
Solution AB
2 2 2
A
1
0
4
0 4
B 2 7
1 3
A is a 2 3 matrix and B is a 3 2 matrix, so AB
will be a 2 2 matrix.
0
2 2 2
AB
2
1 0 4 1
2(0) 2(2) 2(1)
1(0) 0(2) 4(1)
4
7
3
2( 4) 2( 7) ( 2)(3) 6 28
1(4) 0(7) 4(3) 4 8
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 16
Solution BA
2 2 2
A
1
0
4
0 4
B 2 7
1 3
B is a 3 2 matrix and A is a 2 3 matrix, so BA
will be a 3 3 matrix.
0 4
2
BA 2 7
1
1 3
(0)(2) (4)(1)
(2)(2) (7)(1)
(1)(2) 3(1)
2 2
0 4
0(2) (4)(0) (0)(2) (4)(4) 4 0 16
2(2) (7)(0) (2)( 2) ( 7)(4) 3 4 32
1(2) 3(0) (1)(2) (3)(4) 1 2 14
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 17
Solution AC
2 2 2
A
1
0
4
2 4
C
1
0
The product AC is not defined because the number
of columns of A, 3, is not equal to the number
of rows of C, 2.
• Note that AB BA. Multiplication of matrices is
generally not commutative.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 18
Application
Dalton’s Dairy produces no-fat ice cream and frozen yogurt.
The following table shows the number of gallons of each
product that are sold at the dairy’s three retail outlets one
week. On each gallon of no-fat ice cream, the dairy’s profit is
$4, and on each gallon of frozen yogurt, it is $3. Use matrices
to find the total profit on these items at each store for the
given week.
Store 1
Store 2
Store 3
No-fat Ice Cream (in
gallons)
100
80
120
Frozen Yogurt (in
gallons)
160
120
100
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Slide 6.4 - 19
Application continued
We can write the table showing the distribution as a
2 3 matrix.
100 80 120
D
160
120
100
The profit per gallon can also be written as a matrix.
P 4 3
The total profit at each store is given by the matrix
product PD.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 20
Application continued
100 80 120
PD 4 3
160
120
100
4(100) 3(160) 4(80) 3(120) 4(120) 3(100)
880 680 780
The total profit on no-fat ice cream and frozen
yogurt for the given week was $880 at store 1, $680
at store 2, and $780 at store 3.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 21
Properties of Matrix Multiplication
For matrices A, B, and C, assuming that the
indicated operations are possible:
Associative Property of Multiplication
A(BC) = (AB)C.
Distributive Property
A(B + C) = AB + AC.
(B + C)A = BA + CA.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide 6.4 - 22
Matrix Equations
We can write a matrix equation equivalent to a
system of equations.
Example: 3 x y z 11
5x
z 9
x 2 y 3 z 3
1 1 x 11
5 0 1 y 9
1 2 3 z 3
Can be written as: 3
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Slide 6.4 - 23
Matrix Equations
If we let
3 1 1
x
11
A 5 0 1 , X y , and B 9
1 2 3
z
3
We can write this matrix equation as AX = B.
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Slide 6.4 - 24