Admittedly, Matrix multiplication is a cleanly defined concept. However, it’s rather non-intuitive and non-visual to many people. There are quite a few “rules of thumb” about it, but many of them are hard to internalize due to the abstract nature. They are not intuitive enough to “take root” in our mind.
I find it effective to focus on a few simple, intuitive rules and try to internalize just 1 at a time.
Rule — a 2×9 * 9×1 is possible because the two “inside dimensions” match (out of the 4 numbers).
Rule — in many multiplication scenarios, you can divide-and-conquer the computation process BY-COLUMN — A vague slogan to some students. It means “work out the output matrix column by column”. It turns out that you can simply split a 5-column RHS matrix into exactly 5 columnar matrices. Columnar 2 (in the RHS matrix) is solely responsible for Column 2 in the output matrix. All other RHS columns don’t matter. Also RHS Column 2 doesn’t affect any other output columns.
You may be tempted to try “by-row”. I don’t know if it is valid, but it’s not widely used.
By-column is useful when you represent 5 linear equations of 5 unknowns. In this case, the RHS matrix comprises just one column.
Rule — Using Dimension 3 as an example,
(3×3 square matrix) * (one-column matrix) = (another one-column matrix). Very common pattern.