Properties of Matrix Multiplication

Matrix multiplication is associative.

\[(\M{A}\cdot \M{B})\cdot \M{C}\quad =\quad \M{A}\cdot (\M{B}\cdot \M{C})\]

Intuitively, the “order of application” is right-to-left.

\[\M{A}\cdot \M{B}\cdot \M{C}\cdot \V{v}\quad =\quad \M{A}\cdot (\M{B}\cdot (\M{C}\cdot \V{v}))\] \[\M{A}\cdot \M{B}\cdot \M{C}\cdot \V{v}\quad =\quad ((\M{A}\cdot \M{B})\cdot \M{C})\cdot \V{v}\] \[\M{A}\cdot \M{B}\cdot \M{C}\cdot \V{v}\quad\neq\quad \M{C}\cdot (\M{B}\cdot (\M{A}\cdot \V{v}))\]