10.2 Multiplying matrices i

Just to limber up your matrix muscles, let’s try doing the following matrix addition.

This brings us to the world of multiplying a matrix by a number. It’s very straightforward. You end up with a matrix that has the same dimensions as the original, but all the individual cells have been multiplied by that number.

Let’s do another example. I’m sure you remember Professor Snape’s grade matrix.

Now, we saw how Professor Snape could lower his grades (which he loves to do) by subtracting a curve matrix . But there is another way he can lower his grades, which is by multiplying the entire matrix by a number. In this case, he is going to multiply his grade matrix by $\frac{9}{\text{10}}$ . If we designate his grade matrix as $\left[S\right]$ then the resulting matrix could be written as $\frac{9}{\text{10}}$ $\left[S\right]$ .

Finally, it’s time to Prof. Snape to calculate final grades. He does this according to the following formula: “Poison” counts 30%, “Cure” counts 20%, “Love philter” counts 15%, and the big final project on “Invulnerability” counts 35%. For instance, to calculate the final grade for “Granger, H” he does the following calculation: $(30\%)\left(100\right)+(20\%)\left(105\right)+(15\%)\left(99\right)+(35\%)\left(100\right)=100.85$ .

To make the calculations easier to keep track of, the Professor represents the various weights in his grading matrix which looks like the following:

$\left[\begin{array}{c}\text{.}3\\ \text{.}2\\ \text{.}\text{15}\\ \text{.}\text{35}\end{array}\right]$

The above calculation can be written very concisely as multiplying a row matrix by a column matrix, as follows.

$\left[\begin{array}{cccc}\text{100}& \text{105}& \text{99}& \text{100}\end{array}\right]$ $\left[\begin{array}{c}\text{.}3\\ \text{.}2\\ \text{.}\text{15}\\ \text{.}\text{35}\end{array}\right]=\left[100.85\right]$

A “row matrix” means a matrix that is just one row. A “column matrix” means…well, you get the idea. When a row matrix and a column matrix have the same number of items, you can multiply the two matrices. What you do is, you multiply both of the first numbers, and you multiply both of the second numbers, and so on…and you add all those numbers to get one big number. The final answer is not just a number—it is a 1×1 matrix, with that one big number inside it.