I agree. Matrix multiplication can pretty succinctly be defined in terms of the dot product between the row and column of the inputs at the corresponding spot i

I agree. Matrix multiplication can pretty succinctly be defined in terms of the dot product between the row and column of the inputs at the corresponding spot in the output. When I first learned I already new a little about the properties of dot products so it made the “magical” properties of matrix multiplication more understandable. Maybe not the way a mathematician would define it — they might look at dot products as a special case of matrix multiplication thus teach the latter first — but it was at least intuitive.