How do you multiply matrices? I realized that the technique that I learned back in the days is not as common as I thought. I often see my students struggling to perform simple matrix multiplications, and I would argue that part of the reason is that they write them *side-by-side*. This is also what I see in the books I could get my hands on, and in some popular videos online.

To perform the multiplication of two matrices, I learned to write things in the way depicted in the picture below.

The advantages are multiple. First, to compute the element in position of , the definition of matrix multiplication says that we need the -th row of and the -th column of , which we can neatly see at the left and the top of position .

Therefore, doing the operation requires only a minimum of mental gymnastics and eye movement.

Added advantage:the matrix multiplication formula

,

is right there under your eyes, no need to remember it. Provided of course that we remember that an element is in the -th row and -th column, not vice versa, but this is convention and we cannot do anything about it.

A third selling point of this method is that it makes it easy to see the size of matrices.

- We can multiply and if we can “lay down” B on A after rotating it . In other words, the top part of and left part of can be glued together perfectly. In symbols, we can multiply a matrix by a matrix.

- The matrix multiplication fits snuggly in the space left on the right of and the bottom of . In other words, the product of a matrix by a matrix is a matrix.

As far as I am concerned, this is the only way I can recover all these formulas.

Consider matrices of size respectively . Under what conditions is the matrix product well-defined? What is the size of the resulting matrix?

Consider a square matrix , and the column vector of size that is filled with . Explain in words what , and are. The exponent is the transpose.