College Mathematics: Learning Worksheets Chapter 4
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14. A dog food manufacturer makes two types of dog food: Sparky’s Special Selection and
Fido’s Favorite Food. Each type of food uses two types of additives: Additive #1 and
Additive #2. Each bag of Sparky’s Special Selection has 7 units of Additive #1 and 6 units
of Additive #2. Each bag of Fido’s Favorite Food has 9 units of Additive #1 and 4 units of
Additive #2. On one day last month the dog food manufacturer had 209 units of Additive #1
and 142 units of Additive #2 available. How many bags of each type of dog food could be
made to use exactly the amount of additives available?
Now solve the above equations.
124
15. A dog food manufacturer makes two types of dog food: Sparky’s Special Selection and
Fido’s Favorite Food. Each type of food uses two types of additives: Additive #1 and
Additive #2. Each bag of Sparky’s Special Selection has 5 units of Additive #1 and 3 units
of Additive #2. Each bag of Fido’s Favorite Food has 2 units of Additive #1 and 4 units of
Additive #2. On one day last month the dog food manufacturer had 151 units of Additive #1
and 155 units of Additive #2 available. How many bags of each type of dog food could be
made to use exactly the amount of additives available?
Let f be the number of bags of Fido’s Favorite and s be the number of bags of Sparky’s
Special. Based on the amount of available additives, we would have the following:
Now solve the above equations.
College Mathematics: Learning Worksheets Chapter 4
Name ________________________________ Date ______________ Class ____________
Goal: To use the basic operations (add, subtract, and multiply) on matrices
Perform the indicated operations, if possible.
1.
739 83 4
72 4 2 1 2
−
⎡⎤⎡⎤
⎢⎥⎢⎥
−+−−
=
15 6 5
51 2
⎡
⎤
⎢
⎥
−
⎢
⎢
⎣
2. 56 56
−−
⎡⎤⎡⎤
+
⎢⎥⎢⎥
=00
Section 4-4 Matrices: Basic Operations
Matrix Addition (Subtraction):
ab wx awbx
cd yz cydz
Multiplication of a nonzero constant:
Matrix Multiplication:
Given two matrices, the first is an aband the second is ,cdthe product is defined
College Mathematics: Learning Worksheets Chapter 4
131
9 5 11 10 5 2
−
⎡⎤⎡ ⎤
1013
⎡⎤
⎡⎤
−
⎡⎤
−−
⎡⎤
9. 1517
−
⎡⎤⎡⎤
⋅
⎢⎥⎢⎥
=(1)(1) ( 5)(4) (1)(7) ( 5)( 2) 19 17
+− +− − −
⎡⎤⎡⎤
=
⎢⎥⎢⎥
College Mathematics: Learning Worksheets Chapter 4
127
10.
81 3 1 2 3
=
( 8)(7) (1)(1) (3)(6) ( 8)(2) (1)( 2) (3)(9) ( 8)( 5) (1)( 3) (3)( 1)
(4)(7) (4)(1) (8)(6) (4)(2) (4)(2) (8)(9) (4)(5) (4)(3) (8)(1
)
92 80 44
11. 41 15
34
52 2 4
12.
12 3
04 5
=
13
20
13.
12
304
551
The first matrix is a 2 1and the second matrix is a 3 2,therefore,
14. 2301
5410
−
⎡⎤⎡⎤
⋅
⎢⎥⎢⎥
−
⎣⎦⎣⎦
=32
45
−
⎡⎤
⎢⎥
−
⎣⎦
College Mathematics: Learning Worksheets Chapter 4
128
College Mathematics: Learning Worksheets Chapter 4
129
Name ________________________________ Date ______________ Class ____________
Goal: To find inverses of square matrices
Find the inverse of the following matrices. If the matrix does not have an inverse,
explain why.
1. 614
24
−
⎡⎤
⎢⎥
−
⎣⎦
1
db
−
−
⎡
⎤
⎢
⎣
⎡
⎢
⎣
Section 4-5 Inverse of a Square Matrix
To find the inverse of a matrix:
a) form the augmented matrix
M
I
I
If we obtain all zeros in one or more rows to the left of the vertical line, the 1
M
does
not exist.
The inverse of a 2 2 matrix can also be found by using the following formula:
2. 64
85
⎢⎥
⎣⎦
1
db
−
−
⎡
⎤
⎢
⎣
⎡
⎢
⎣
⎡
⎢
⎢
⎣
3. 21
32
⎡⎤
⎢⎥
⎣⎦
db
−
4. 22
51
1
db
College Mathematics: Learning Worksheets Chapter 4
131
5. 11
32
1
db
6.
12 4
42
⎡⎤
⎢⎥
−
7. 23
53
8.
9
College Mathematics: Learning Worksheets Chapter 4
132
9.
10 1
21 0
01 1
10.
012
112
314
012100 1 12010
23 3
4
01 2100 01 21 00
RR R
0.6 0.2 0.4
College Mathematics: Learning Worksheets Chapter 4
Name ________________________________ Date ______________ Class ____________
Goal: To solve systems of linear equations using inverse matrices
In Problems 1–4, write the system as a matrix equation of the form AX B.
1. 12
83 14
xx
+=
⎧
⎨+=
12
42 8
xx
3. 12
12
37 65
23
xx
xx
+=
⎧
⎨+=
⎩ 4.
12
12
25 31
23 25
xx
xx
Section 4-6 Matrix Equations and Systems of
Linear Equations
Using Inverse Methods to Solve Systems of Equations:
If the number of equations in a system equals the number of variables and the
coefficient matrix has an inverse, then the system will always have a unique solution that
can be found by using the inverse of the coefficient matrix to solve the corresponding
matrix equation:
College Mathematics: Learning Worksheets Chapter 4
134
5. 1
2
83 14
35 13
x
x
⎡⎤
⎡⎤ ⎡⎤
=
⎢⎥
⎢⎥ ⎢⎥
⎣⎦ ⎣⎦
⎣⎦
6. 1
2
42 8
53 4
x
x
1.5 1