Why is it that when multiplying a (1x3) by (3x1) matrix, you get a (1x1) matrix, but when multiplying a (3x1) matrix by a (1x3) matrix, you get a (3x3) matrix? Why is matrix multiplication defined this way?
Why can't a (1x3) by (3x1) yield a (3x3), or a (3x1) by (1x3) yield a (1x1)? I really would like to get to the root of this problem or 'axiomatization'. Thanks.