Proof of Theorem MMSMM

These are equalities of matrices. We will do the first one, the second is similar and will be good practice for you. For $1\leq i\leq m$, $1\leq j\leq p$, \begin{align*} \matrixentry{\alpha(AB)}{ij} &=\alpha\matrixentry{AB}{ij}&&\knowl{./knowls/definition.MSM.knowl}{\text{Definition MSM}}\\ &=\alpha\sum_{k=1}^{n}\matrixentry{A}{ik}\matrixentry{B}{kj}&&\knowl{./knowls/theorem.EMP.knowl}{\text{Theorem EMP}}\\ &=\sum_{k=1}^{n}\alpha\matrixentry{A}{ik}\matrixentry{B}{kj}&&\knowl{./knowls/property.DCN.knowl}{\text{Property DCN}}\\ &=\sum_{k=1}^{n}\matrixentry{\alpha A}{ik}\matrixentry{B}{kj}&&\knowl{./knowls/definition.MSM.knowl}{\text{Definition MSM}}\\ &=\matrixentry{(\alpha A)B}{ij}&&\knowl{./knowls/theorem.EMP.knowl}{\text{Theorem EMP}} \end{align*}

So the matrices $\alpha(AB)$ and $(\alpha A)B$ are equal, entry-by-entry, and by the definition of matrix equality (Definition ME) we can say they are equal matrices.