Proof of Theorem MMIM

Again, we will prove (1) and leave (2) to you. Entry-by-entry, For $1\leq i\leq m$, $1\leq j\leq n$, \begin{align*} \matrixentry{AI_n}{ij}=& \sum_{k=1}^{n}\matrixentry{A}{ik}\matrixentry{I_n}{kj} &&\knowl{./knowls/theorem.EMP.knowl}{\text{Theorem EMP}}\\ &=\matrixentry{A}{ij}\matrixentry{I_n}{jj}+\sum_{\substack{k=1\\k\neq j}}^{n}\matrixentry{A}{ik}\matrixentry{I_n}{kj} &&\knowl{./knowls/property.CACN.knowl}{\text{Property CACN}}\\ &=\matrixentry{A}{ij}(1)+\sum_{k=1, k\neq j}^{n}\matrixentry{A}{ik}(0) &&\knowl{./knowls/definition.IM.knowl}{\text{Definition IM}}\\ &=\matrixentry{A}{ij}+\sum_{k=1, k\neq j}^{n}0\\ &=\matrixentry{A}{ij} \end{align*}

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