Question

A music recording company has contracts with two bands. The CDs of these bands are released in three countries, Britain, Canada, and Australia. The total number of CDs sold in each country is Britain, 40,000; Canada, 50,000; and Australia, 30,000. The cost of advertising for both bands was $2 per CD in Britain, $1.50 in Canada, and $1 in Australia. If matrix A = [2 1.5 1], what is matrix B if AB represents the total advertising cost (in dollars) for all three countries?

Answer 1

In this problem we are asked to find the total advertising cost of the CDs for three countries.  This can be represented by a multiplication of matrices.

A is an 1x3 matrix and represents The cost of advertising for both bands

B must be a 3x1 matrix and represents The total number of CDs sold in each country

So, according to the multiplication of matrix rules, the result of multiplying a 1x3 matrix by a 3x1 matrix is an 1x1 matrix.

Let's begin:

$A=\left[\begin{array}{ccc}2&1.5&1\end{array}\right]$

$B=\left[\begin{array}{ccc}40000\\50000\\30000\end{array}\right]$

The correct way to multiply these matrices is as follows:

$AB=\left[\begin{array}{ccc}2&1.5&1\end{array}\right]\left[\begin{array}{c}40000\\50000&30000&\end{array}\right]$

Then:

AB=[2(40000)+1.5(50000)+1(30000)]

AB=[80000+75000+30000]

Finally:

AB=[185000]

The answer is $185000, which is the total advertising cost for all three countries