3.2 Matrix Multiplication

3.2 Matrix Multiplication

Question 1: How do you multiply two matrices?

Question 2: How do you interpret the entries in a product of two matrices?

When you add or subtract two matrices, you add or subtract the entries in two matrices of the same size. You might try to multiply two matrices by following a similar strategy.

However, matrix multiplication is not carried out by multiplying the corresponding entries of two matrices of the same size.

Instead, matrix multiplication is carried out by multiplying the entries in the rows of a matrix by the entries in the columns of the other matrix. This might not seem to be a productive process. However, this process is very useful in many areas of business, economics, and science.

In this section you’ll learn how to carry out this process and apply it to several problems at Ed Magazine.

Question 1: How do you multiply two matrices?

The process of multiplying matrices is different from scalar multiplication or the other matrix operations in the previous section. Instead of multiplying corresponding entries, in matrix multiplication we multiply the rows in one matrix by the columns in the another matrix. This process can be demonstrated by multiplying a row matrix times a column matrix. Suppose we have a 1 x k matrix,





A a a  a

and a k x 1 matrix,

11 21

1 n

b B b

b

  

 

  

 



In each matrix, the dots help to indicate the arbitrary number of rows or columns in each matrix. Although this number k is arbitrary, the number of columns in A must match the number of rows in B. Otherwise it is not possible to carry out the multiplication process.

To find the product these matrices, we must multiply the entries in the row matrix by the entries in the column matrix and add the resulting products:

 

11

11

11 11

21 12

1

1

1

1

2 1

2 1

n

n

n n

a b

a b a b

a

b b

B

b

A a

a

  

  

  

 

   

 



Notice that each product comes from corresponding columns and rows. In other words, the first product is formed from the first column in the first matrix and the first row in the second matrix, the second product is formed from the second column in the first matrix and the second column in the second matrix, and so on.

Let’s try the following product:

 

2 3 1 2 4 9

2 1

  

  

 

  

To help identify the factors in the products, let’s color code each corresponding factor and carry out the sum:



       

 

2 2

3 3 1 5

1

1 9 2

1 9

  

       



  



 



The key to carrying out the process is to correspond the factors in each product correctly.

This process is carried out several times when matrices with more than one row or column are multiplied. However, the number of columns in the first matrix must match the number of rows in the second matrix.

How to Multiply Two Matrices

1. Make sure the number of columns in the first matrix matches the number of rows in the second column. If they do not match, the product is not possible.

2. The size of the products is the number of rows in the first matrix by the number of columns in the second matrix.

The product of m x k matrix and a k x n matrix is an m x n matrix. Form a matrix of the proper size with blank spaces for each entry.

3. For each entry in the product, form the corresponding factors and sums. The entry in the i^th row and j^th column of the product is found by corresponding and multiplying the i^th row in the first matrix with the j^th column in the second matrix.

Example 1 Multiply Two Matrices

Let

1 1 0

2 1 3

A   

  

   and   2 4

B  1 3

   

Find the products indicated in each part.

a. A B

Solution To be able to compute this product, the number of columns in A must equal the number of rows in B. Since A has 3 columns and B has 2 rows,

 

2

23 2

A B

 

,

it is not possible to compute this product.

b. B A

Solution For this product, the number of columns in B is equal to the number of row in A,

 

2

2 2 3

B A

 

Not Equal

Equal

This means the product can be computed. The size of the resulting product is determined by the number of rows in B, 2, and the number of columns in A,3:

 

2

2 2 3

B A

 

Now that the size of the product is known, we can find the entries in the product.

Start with blank entries in a 2 x 3 matrix:

 

 

 

We can find the value of any entry in the product by corresponding the proper row and column in the factors. For instance, the entry in the second row, first column is computed from the second row of the first matrix and the first column of the second matrix:

3 1 1

2 4 1 0

3 2 1

BA     

   

   

 ^{1 1}

 

 

This entry is placed in the product matrix,

5

 

 

 

The entry in the first row, third column is computed from the first row of the first matrix and the third column of second column:

Product is 2 x 3

1 1

1 3 2 1

2 4 0

BA     3

      ^{2 0}

 

 

Adding this entry to the product matrix yields 12 5

 

 

 

We can compute the other four entries in the product matrix similarly.

Example 2 Multiply Two Matrices

The table below gives the number of expiring subscriptions for Ed Magazine.

First Time Subscribers Continuing Subscribers

Quarter Ending 3/31 6000 15000 Quarter Ending 6/30 2000 2600 Quarter Ending 9/30 6500 12000 Quarter Ending 12/31 1500 3600

This information is summarized in the matrix

2 4 1 1 0 10 2 12

1 3 2 1 3 5 4 9

BA       

       

   

2 1 4 2 10 ²

   

   

1 1 3 1 4

    ^{1 0}

   

6000 15000 2000 2600 6500 12000 1500 3600 E

 

 

 

 

 

 

The different categories of subscribers renew their subscriptions at different rates. Twenty five percent of the first time subscribers renew their subscriptions and fifty percent of the existing subscribers renew their subscriptions.

a. Use matrix multiplication to find a matrix describing the total number of renewed subscribers by quarter.

Solution To see how matrix multiplication can be used to calculate the total number of renewed subscribers, watch the video, let’s look at the quarter ending 3/31. In that quarter, 6000 first time subscribers and 15000 continuing subscribers have their subscriptions expiring. We know that 25% of the first time subscribers will renew and 50% of the continuing subscribers will renew.

The total number of renewed subscriptions in the first quarter is

   

25% of FirstTime Subscribers 50% of Continuing Subscribers

Total Number of Renewed

0.25 6000 0.50 15000 9000

Subscriptions in First Quarter  

 

We can also calculate the total number of renewed subscriptions in other quarters using this same strategy.

   

   

Total Number of Renewed

2000 0.25 2600 0.50 1800 Subscriptions in Second Quarter

Total Number of Renewed

6500 0.25 12000 0.50 7625 Subscriptions in Third Quarter

Total Number of Renewed Subscriptions in Fourth

  

  

   

1500 0.25 3600 0.50 2175

Quarter  

Notice that each number is the sum of two products. The product of two matrices creates a new matrix where each entry is a sum of products.

This suggests that we define a matrix

Renewal Rate

First Time Subscribers Continuing Subscribers

0.25 P 0.50

  

 

of the renewal rates for the subscribers groups.

The product

can be carried out since E has 2 columns and P has 2 rows.

The resulting product is a 4 x 1 matrix:

Notice that each entry matches the totals found earlier. Using matrices we are able to compute the total number of renewals by quarter

efficiently. Additionally, if more quarters are included in E the process

can still be carried out by adding more rows to E.

9000 1800 7625 2175 EP

 

 

 

 

 

 

   

6000 0.25 15000 0.50 9000

   

2000 0.25 2600 0.50 1800

   

6500 0.25 12000 0.50 7625

   

1500 0.25 3600 0.50 2175 4 x 2 2 x 1

6000 15000 2000 2600 0.25 6500 12000 0.50 1500 3600

E P

 

   

 

   

 

 

b. A renewing subscriber pays $18 per year for a subscription. Find a matrix R that gives the cash receipts from renewed subscriptions ₂ by quarter.

Solution The product EPgives the total number of renewed subscriptions by quarter. To find the cash receipts from these subscriptions, we must multiply each entry in the product by 18.

Multiplying the product EP by the scalar 18 gives

2 18

9000 18 1800

7625 2175 162000

32400 137250

39150 R  EP

 

 

 

  

 

 

 

 

 

 

 

 

c. The matrix

1

52000 30000 56000 25000 R

 

 

 

 

 

 

gives the cash receipts from new subscriptions by quarter. Find the matrix R that gives the total cash receipts from new and existing subscriptions.

Solution The total cash receipts R is the sum of cash receipts from new subscriptions R and cash receipts from existing subscriptions ₁ R , ₂

Replace EP with the product from part a

Multiply each entry by 18

1 2

R  R R

Combine R and ₁ R to yield ₂ 52000 162000 30000 32400 56000 137250 25000 39150 214000

62400 193250

64150 R

   

   

   

 

   

   

   

 

 

 

 

 

 

Add corresponding entries in each matrix Replace R₁ and R₂ with matrices

Question 2: How do you interpret the entries in a product of two matrices?

Before attempting to compute or interpret what the product tells you, it is instructive to determine the size of the product. As indicated earlier, the product of an m x k matrix and a k x n matrix is an m x n matrix. Once we know the size of the product, we can compute each of the entries in the product. The entries in the product are formed by corresponding the rows and columns in the factors, multiplying the entries, and

summing the results. This operation is often very useful in computing various quantities in business. However, it is often not obvious exactly what the product tells you.

In a typical application, we can use the labels on the number of rows m in the first matrix to label the rows of the product. To label the columns in the product, write out the

calculation for the first entry with the units on each factor. By analyzing the units, we can deduce what that entry tells us. The other entries will have a similar interpretation to the first entry.

Example 3 Interpret the Product of Two Matrices

The number of new subscriptions by quarter is given by the matrix

Service Magazine

Quarter Ending 3/31 Quarter Ending 6/30 Quarter Ending 9/30 Quarter Ending 12/31

2800 2400 4200 4000 5000 8800 8000 8300 N

 

 

 

 

 

 

New subscriptions may come from a subscription service or may come from the magazine’s marketing. The columns of N indicate the number of subscriptions from each source.

Find and interpret the product

2800 2400 4200 4000 1 5000 8800 1 8000 8300

 

   

   

   

 

 

Solution In this product we are multiplying a 4 x 2 matrix times a 2 x 1 matrix. Since the number of columns (2) in the first matrix matches the number of rows in the second matrix (2), we can carry out the matrix multiplication. The resulting product will be a 4 x 1 matrix:

Notice that each entry in the product is simply the sum of the entries on the same row in the first matrix. Since these values are the number of new subscriptions in that quarter, the sum in the product corresponds to the total number of new subscriptions in that quarter.

For instance, in the first quarter a total of 5200 new subscriptions were received from the subscription service and the magazine’s marketing efforts,



  

 

The numbers in the second matrix have no units. The effect of multiplying by the matrix 1

1

  

  is to add the entries in each row of the matrix N.

   

2800 1 2400 1 5200

   

4200 1 4000 1 8200

   

5000 1 8800 1 13800

   

8000 1 8300 1 16300

2800 2400 5200

4200 4000 1 8200 5000 8800 1 13800

8000 8300 16300

   

     

    

     

   

   

Example 4 Interpret the Product of Two Matrices

The new subscriptions described by the matrix

Service Magazine

Quarter Ending 3/31 Quarter Ending 6/30 Quarter Ending 9/30 Quarter Ending 12/31

2800 2400 4200 4000 5000 8800 8000 8300 N

 

 

 

 

 

 

contribute different amounts of cash to Ed Magazine. Subscriptions enlisted by the subscription service pay $10 for a subscription, but only

$2 goes to the magazine. Subscriptions developed through the

magazine’s marketing campaigns pay $12 and all of this cash goes to the magazine. We can summarize this information in the matrix

Dollars per subscription

Service Magazine

2 S  12

  

 

Find and interpret

2800 2400 4200 4000 2 5000 8800 12 8000 8300 NS

 

   

 

    

 

 

Solution Let’s check the size of each matrix to insure that the matrix multiplication is possible.

The number of columns in N representing the number of new

subscriptions and the number of rows in S representing the cash from subscriptions are both equal to 2 so the multiplication can be carried out to give a 4 x 1 product.

We can form the entries in the product by corresponding the rows in N with the column in S:

The four rows in the product correspond to the four quarters, but what do the entries tell us about those quarters?

To answer this question, let’s look at the first entry in detail:

subscript

2800 ions

 

subsc doll

rip 2 ars

tion 2400 subscriptions

 

 

 

 

 

subs do

cr llar

ip 12 s

tion

 

 

 

 

Each term has units of dollars and indicates the amount of cash received from the sales of subscriptions to new subscribers of each type (from the subscription service and from the magazine’s

2800 2400 34400

4200 4000 2 56400 5000 8800 12 115600

8000 8300 115600

NS

   

     

   

     

   

   

   

2800 2 2400 12 34400

   

4200 2 4000 12 56400

   

5000 2 8800 12 115600

   

8000 2 8300 12 115600 2800 2400

4200 4000 2

5000 8800 12 8000 8300

NS

 

   

 

    

 

 

quarters x categories of number of new subscriptions categories of number of new subscriptions x price (4 x 2) (2 x 1)

promotions). So the sum, 34400 dollars, represents the total amount of cash received from both types of subscribers together.

Other entries can be analyzed similarly to show the total cash received from new subscribers in the other three quarters.