### Video Transcript

The table shows the number of different types of rooms in three hotels owned by a company. If a single room costs 160 Egyptian pounds per night, a double room costs 430 Egyptian pounds per night, and a suite costs 740 Egyptian pounds per night, determine the company’s daily income when all the rooms are occupied.

We can answer this question using matrices. The table containing the number of different types of rooms in the three hotels can be thought of as a three-by-three matrix. In order to find the company’s daily income, we need to multiply the matrix containing the number of rooms by a matrix containing the costs of each type of room.

In order to be able to multiply these two matrices together, the number of rows in the second matrix, the matrix of costs, needs to be the same as the number of columns in the first matrix. Therefore, the matrix of costs needs to be a three-by-one column matrix. The elements for this matrix are the costs for the three different types of room: 160 Egyptian pounds for the single room, 430 Egyptian pounds for the double room, and 740 Egyptian pounds for a suite.

Now we multiply the two matrices together, remembering the process for multiplying matrices. As we multiply a matrix of order three by three by a matrix of order three by one, the product will be a matrix of order three by one, so another column matrix with three rows.

First, we multiply the elements in the first row of the first matrix by the elements in the first and in this case only column in the second matrix. This gives 45 multiplied by 160 plus 74 multiplied by 430 plus 15 multiplied by 740, which is 50120. So we found the first element in our product matrix.

To find the second element, we multiply the second row of the first matrix by the first and only column of the second. This gives 48 multiplied by 160 plus 74 multiplied by 430 plus 19 multiplied by 740, the total of which is 53560.

To find the final element, we multiply the third row of the first matrix by the first and only column of the second. This gives 49 multiplied by 160 plus 94 multiplied by 430 plus 10 multiplied by 740, the total of which is 55660.

So now that we found the product of these two matrices, what does each element in the product represent? The elements in the product represent the total cost for each hotel. The total cost for the first hotel for all the singles, doubles, and suites is 50120 Egyptian pounds. For the second hotel, it’s 53560 Egyptian pounds, and for the third 55660 Egyptian pounds.

To find the total daily income for the company, we need to add all three elements of this matrix together. The sum of these three values is 159340. So this is the total daily income for the company.

Now there is actually a second approach that we could take to answering this question. Each type of room cost the same regardless of which hotel it’s in. It doesn’t actually matter whether it’s in the first hotel, the second, or the third. Therefore, we could condense the first matrix into a row matrix by adding together the number of single rooms in each hotel, the number of double rooms in each hotel, and the number of suites. This would give the row matrix 142, 242, 44.

We’d now be multiplying a one-by-three matrix by a three-by-one matrix to give a one-by-one matrix, that is, the overall answer. The calculation would be 142 multiplied by 160 plus 242 multiplied by 430 plus 44 multiplied by 740. And you can confirm that this gives the same answer as before, 159340.

In the first approach to this question, we had to do some summing for the three different hotels at the end, whereas in the second approach, we summed across the three different hotels at the start. Both approaches are equally valid and give the same result, 159340 Egyptian pounds.