How do you multiply a matrix by a scalar?
In mathematics, real numbers are often called scalars. The process of multiplying a matrix by a scalar is simply a matter of multiplying each entry in the matrix by the scalar.
The product of a scalar c and a matrix
is
For instance, the circulation of issues to new subscribers at Ed Magazine is given by the column matrix
In this matrix, the rows correspond to the different quarters. This means the entry in the second row, first column tells us that in the second quarter 8200 issues of Ed Magazine was distributed to new subscribers. If each issues costs $0.60 to print and mail, the product
gives the expense of printing and mailing the magazine to new subscribers in each quarter. We can calculate the expense of printing and mailing the magazine to other subscribers similarly.
Example 4 Multiply a Matrix by a Scalar
It costs Ed Magazine $0.60 to print and mail one issue of the magazine. The number of issues distributed to renewing subscribers is given by the matrix C2 and the number of issues distributed to non-renewing subscribers is given by the matrix C3 where
The rows of each matrix refer to the first, second, third and fourth quarters of the year.
a. Find the product 0.60 C2.
Solution To multiply the circulation matrix by the scalar 0.60, multiply each entry in the matrix by 0.60:
b. Find the product 0.60 C3.
Solution Multiply each entry in the circulation by the cost per issue 0.60 to give
c. Interpret what the products in parts a and b mean.
Solution Each entry in the product comes from multiplying the expense of printing and mailing a single issue in dollars per issue by the number of issues printed and sent to renewing subscribers. In the product 0.60 C2, each entry is
This is the expense of printing and mailing all issues to the renewing subscribers in the quarter corresponding to the numbers. Since these subscribers are subscribed for the entire year, the expense is the same through all quarters. For the product
each entry in the product yields the expense to print and mail issue of the magazine to non-renewing subscribers in each quarter. Note that this expense drops throughout the year as the number of non-renewing subscribers drops off.