The way things are you have a nice set of properties which allow you to manipulate system of matrix equations - but it's just notation and convention. You could

The way things are you have a nice set of properties which allow you to manipulate system of matrix equations - but it's just notation and convention. You could do column by row as well - it'd be all the same math just effectively transposed. If you defined matrix multiplication as row times row then you are either limited to square matrices or you loose some of the convenient properties of matrix multiplication ABC = A(BC) = (AB)C (AB)^T = B^T A^T and most importantly AB =/= BA