MATH 141-501
Section 2.5 Lecture Notes
Interpreting and Multiplying Matrices
Summary of Matrices and Linear Systems
rref form
Matrix
of
Augmented
Example
1 0 0
 0 1 0
0 0 1

2
4 
3
Exactly one solution (unique)


2
4 
3
No solutions

Coefficient matrix has all unit
columns
Coefficient matrix has a row of
all zeros, but last entry of matrix is not zero!
Number of Solutions
1 0 0
 0 1 0
0 0 0
With any other form (including row of all zeros, we must go back to the
original system of equations and see.
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Interpreting Matrices
Last time, we introduced matrices which were not augmented.
Careful: We only use augmented matrices for solving systems of linear
equations.
In general, we can use matrices
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Example
Example: HardwareCo and ToolboxCo are two competing home supply
and hardware stores in a small town. HardwareCo’s inventory is:
Hardware: Hammers, 12; Screwdrivers, 19; Drills, 24.
ToolboxCo has the following inventory:
Hardware: Hammers, 11; Screwdrivers, 29; Drills, 14.
• Write a matrix A to represent HardwareCo’s inventory.
• Write a matrix B to represent ToolboxCo’s inventory.
• Find the matrix A + B. What does (A + B)11 represent?
• What does (A + B)12 represent?
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A Common Calculation
HardwareCo has 12 hammers, 19 screwdrivers, and 24 drills. HardwareCo
sells hammers for $10, screwdrivers for $4 and drills for $30. What is the maximum revenue HardwareCo can obtain with its current inventory?
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Multiplying a Row by a Column

Multiply the matrices
12
19
24

10
and  4 .
30

If A =
a11
a12
a13
···
a1n


and B = 

b11
b21
..
.
bn1
AB =
.
What is BA?
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


 is given by

Multiplying Matrices
Let’s do some examples based on problems in Finite Mathematics for the
Managerial Life and Social Sciences.
If A and B are two matrices, it is only possible to find the product AB
if the number of
is equal to the number of
. When this is the case, the (i, j) entry of
(AB) comes from
Suppose A =
1
3
2
0
and B =
1
1
What is AB?
What is BA?
6
0
2
1
−4
.
More Matrix Multiplication
Feel free to use your calculator when doing matrix multiplication.
1 2
1
3
Let C =
and D =
.
0 1
−1 17
What is CD?
What is DC?
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Using and Interpreting Matrix Multiplication
Example: Allison has a collection with 14 Jim Morrison stamps, 11 Bob
Marley stamps, and 3 Charlie Parker stamps. There are two stamp stores in
town. The table below shows how much they pay for each type of stamps.
Store A
Store B
Morrison
$1.55
$1.30
Marley
$0.90
$0.88
Parker
$1.70
$2.32
If Allison is selling her collection, use matrix multiplication to determine
which shop he should sell to.
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Using and Interpreting Matrix Multiplication, part 2
The matrix A gives the percentage of registered voters in a city, classified
by party affiliation and age group.


0.40 0.30 0.30
A =  0.40 0.40 0.20  .
0.45 0.50 0.05
The number of registered voters in the city (by age group) is given by the
matrix B.
30000 40000 25000
1. Find the matrix product BA.
2. Assuming that all registered voters vote, and that they all follow their
party affiliation when voting, which party will win the upcoming mayoral
election?
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