Math 141: Business Mathematics I
Fall 2015
§2.5 Multiplication of Matrices
Instructor: Yeong-Chyuan Chung
Outline
• Another matrix operation - multiplication
• Representing and interpreting data using matrices
Multiplication of Matrices
In this section, we will learn how to multiply a matrix by another matrix. Recall from the
previous section that addition, subtraction, and scalar multiplication are done entrywise.
However, multiplication of matrices is not done entrywise. It turns out that when we are
dealing with data matrices, entrywise multiplication often does not give us the information
we are looking for.
Given two matrices A and B, we write AB for the matrix A × B. To multiply two matrices,
the number of columns in the first (or left) matrix must be the same as the number of rows
in the second (or right) matrix. The resulting matrix will have the same number of rows as
the first matrix, and the same number of columns as the second matrix.
In other words, if A is an m × n matrix, and B is an n × p matrix, then AB is an m × p
matrix.
 
7
1 2 3

Example. If A =
and B = 8, then the product AB is defined because A has
4 5 6
9
3 columns, and B has 3 rows. The matrix AB will have 2 rows and 1 column. On the other
hand, BA is undefined because B has 1 column while A has 2 rows.
From the example, we also see that unlike multiplication of numbers, the order is important
when we multiply matrices. We will see that in general, AB 6= BA even when both AB and
BA are defined. Most of the other multiplication rules for numbers still apply. For example,
A(BC) = (AB)C, and A(B + C) = AB + AC.
1
§2.5 Multiplication of Matrices
2
1 2 3
2 4
Example. Let A =
,B=
, and let C be some unspecified 3 × 2 matrix.
4 5 6
6 8
Which of the following products are defined? For those that are defined, what are their sizes?
1. AB
2. BA
3. A2
4. B 2
5. CA
The entry in the ith row and jth column of AB depends only on the ith row of A and the
jth column of B. The schematic diagram looks something like this:
1 2
3 2 1
Example. Let A =
and B =
. Compute AB.
3 1
6 5 4
§2.5 Multiplication of Matrices
3
Example. Compute the products
−1 2
3 1
Observe that we interchanged the two
1
Example. Compute the products
0
2 4
2 4
−1 2
and
.
3 1
3 1
3 1
matrices, and we got two different products.
0
2 4
2 4
1 0
and
.
1
6 8
6 8
0 1
An n × n matrix with 1’s along the diagonal and 0’s everywhere else (like the one in the
example above) is called the identity matrix. We sometimes write In for this matrix (so the
one above can be written as I2 ).
Recall that in the notes for §2.3, you were told how to enter matrices into your graphing
calculator, and how to call them up after that. So you can use your calculator to compute
sums/differences/products of matrices. (Sometimes, it is faster to just compute by hand.)
§2.5 Multiplication of Matrices
4
Representing and Interpreting Data using Matrices
Example (Exercise 50 in the text). Laura is planning to buy two 5-lb bags of sugar, three
5-lb bags of flour, two 1-gal cartons of milk, and three 1-dozen cartons of large eggs. The
prices of these items in three neighborhood supermarkets are as follows:
Sugar (5-lb bag)
Supermarket I
$3.15
Supermarket II
$2.99
Supermarket III
$3.74
Flour (5-lb bag)
$3.79
$2.89
$2.98
Milk (1-gal carton) Eggs (1-dozen carton)
$2.99
$3.49
$2.79
$3.29
$2.89
$2.99
(a) Write a 3 × 4 matrix A to represent the prices of the items in the three supermarkets.
(b) Write a 4 × 1 matrix B to represent the quantities of the items that Laura plans to
purchase in the three supermarkets.
(c) Find a matrix C, in terms of A and B, that represents Laura’s outlay at each supermarket. At which supermarket should she make her purchase if she wants to minimize
her cost? (Assume that she will shop at only one supermarket.)
§2.5 Multiplication of Matrices
5
Example (Exercise 63 in the text). A dietician plans a meal around three foods. The number
of units of vitamin A, vitamin C, and calcium in each ounce of these foods is represented by
the matrix M , where
The matrices A and B represent the amount of each food (in ounces) consumed by a girl at
two different meals, where
Calculate the following matrices, and explain the meaning of the entries in each matrix.
(a) M AT
(b) M B T
(c) M (A + B)T .