Scalar & Matrix Multiplication

They've given me a matrix and two numbers; namely, the numbers 2 and −1. These two numbers are the scalars. So they're asking me to do scalar multiplication.

To do the first scalar multiplication to find 2A, I just multiply a 2 on every entry in the matrix.

Being excruciatingly complete, the process looks like this:

The other scalar multiplication, to find −1A, works the same way:

I can do the multiplications in my head, of course, but if an instructor wants to see every frickin' step, then I need to write things out like the above. You probably won't need to be this complete in your own work.

Having done the multiplications, my final answer (including which matrix goes with which multiplication) is:

To calculate the product AB, I first write down A and B next to each other like this:

Now I need to multiply the entries in the ROWS of A by the entries in the COLUMNS of B. By this I mean that I take the first row of A and the first column of B, and I multiply the first entries of A's row and B's column, then the second entries of A's row and B's column, and then the third entries of A's row and B's column; and then I add the three products.

The sum of these three multiplications is one entry in the product matrix AB. In fact, being the product of row 1 and column 1, the summed result is the 1,1-entry of the product AB. Then I continue in like manner.

All the entries of the product matrix are found in this way. For instance, the sum of the products from row 2 of A and column 1 of B is the 2,1-entry of AB.

Which is all well and good, but what does this look like in practice?

When I multiply matrices, I use my fingers to keep track of what I'm doing. The following animation is my attempt to illustrate this process. (Don't laugh; I'm no artist!)

(Now, class; what did I say about laughing?) 😜

My final answer is: