Properties of Matrix Multiplication

Matrix multiplication is not commutative.

\[\M{A}\cdot \M{B}\neq \M{B}\cdot \M{A}\]

For example,

\[\M{M} = \M{A}\cdot \M{B}\quad\to\quad m_{0} = a_{0}\cdot b_{0} + a_{4}\cdot b_{1} + a_{8}\cdot b_{2} + a_{12}\cdot b_{3}\]

but,

\[\M{M} = \M{B}\cdot \M{A}\quad\to\quad m_{0} = a_{0}\cdot b_{0} + a_{1}\cdot b_{4} + a_{2}\cdot b_{8} + a_{3}\cdot b_{12}\]