Proof of Theorem MMZM

We will prove (1) and leave (2) to you. Entry-by-entry, for $1\leq i\leq m$, $1\leq j\leq p$, \begin{align*} \matrixentry{A\zeromatrix_{n\times p}}{ij} &=\sum_{k=1}^{n}\matrixentry{A}{ik}\matrixentry{\zeromatrix_{n\times p}}{kj} &&\knowl{./knowls/theorem.EMP.knowl}{\text{Theorem EMP}}\\ &=\sum_{k=1}^{n}\matrixentry{A}{ik}0 &&\knowl{./knowls/definition.ZM.knowl}{\text{Definition ZM}}\\ &=\sum_{k=1}^{n}0\\ &=0 &&\knowl{./knowls/property.ZCN.knowl}{\text{Property ZCN}}\\ &=\matrixentry{\zeromatrix_{m\times p}}{ij} &&\knowl{./knowls/definition.ZM.knowl}{\text{Definition ZM}} \end{align*}

So by the definition of matrix equality (Definition ME), the matrices $A\zeromatrix_{n\times p}$ and $\zeromatrix_{m\times p}$ are equal.