Proposition (Composition rules for $\mathcal{M}$)

Let $V,W,U$ be vector spaces and let $\{v_1,...,v_n\},\{w_1,...,w_m\}$ and $\{u_1,...,u_p\}$ be bases for $V,W,U$ respectively. If $T\in\mathscr{L}(V,W)$ and $S\in\mathscr{L}(W,U)$ then $$\mathcal{M}(S\circ T) = \mathcal{M}(S)\mathcal{M}(T)$$