It's a 50-50 arbitrary convention.

I suggest physically tracing through the matrices with one finger from each hand. As you would multiply If you do it the right way enough times, doing it the wrong way will feel weird.

Also, the canonical example is y=Ax, with A a matrix, and x,y as vertical vectors. If you can at least remember that one goes vertically, and one horizontally, Ax will show that it is the one on the right that goes vertically.