Bra-Ket
The way that vectors and matrices are taught has always struck me as somewhat odd. Even the fact that we think of them as separate concepts is a bit weird, because they are just a particular type of vector. There again, schools do take a good long while to get past the fact that squares are nothing more than a special case of oblong1, so maybe it's a teaching thing in general that is shown to be general better for helping people to learn.
Anyway, the keep point is that I've been dealing with vectors for the better part of 6 years now, and matrices not that much less. However, it's only been in the last few months that I've learnt to reliably remember how to perform matrix multiplication. And, it's done to a notation for vectors, probably most used by physicists, known as Bra-Ket notation.
Leaving all the physical world significance behind, it can be viewed as some simple ideas of how to notate vectors; a row vector is called a bra vector, and a column vector is called a ket vector. This can be remembered by thinking about column or, rather, kolumn vectors.
How to draw them is much easier to remember - they are two halves of a bracket - hence the names 'Bra' and 'Ket' (this is also an 'Ack', but I won't go into that). This notation needs to a couple of nice things. The first is the way that the inner product (dot product) of two vectors is written2; the dot product is the first new concept that is introduced when learning vectors and it has one key property - it is a single value. Now, know that vectors are a form of matrix, and because I know that a product between two matrices will involve reading one row and one column from each cell in the result, I can work out how to multiple matrices.
So, how does all this help? It's because the notation for an inner product is simple and memorable, and this allows me to remember which was round the vectors go. Armed with this knowledge, I now know that each cell uses a row from the first matrix, and a column from the second.