Not sure if anyone here has mentioned this yet, but it's useful to think of matrix multiplication as the most natural extension of one-dimensional algebra into

Not sure if anyone here has mentioned this yet, but it's useful to think of matrix multiplication as the most natural extension of one-dimensional algebra into multiple dimensions. First, remember that people first wrestled with and found solutions to "1-D" equations like "ax^2 + bx + c = 0". When others came along asking about how to extend these results into multiple dimensions so that previous results might still be useful, the most natural answer was found to be how we now define matrix multiplication. To see further evidence of this, take your favorite linear algebraic equation using matrices and then look at the edge case where you have only 1x1 matrices in your equation. Usually you'll end up with something that looks analogous to a result in basic algebra in real number (or complex) variables, not matrices. If all this still seems counter-intuitive, then realize this is essentially why math is so important to our logic. It helps us to reason about things that are not obvious, yet logically must fall through from the basic tautologies we "chose" (like 0 != 1, addition on integer numbers, etc.). Or, more aptly in this case, it helps retain an analogous symbolic representation (the overall logic of the equation at the high level) despite a change in the underlying nuts and bolts of computing solutions to it (matrix multiplication vs. our regular "1D" multiplication).