Proof of Theorem CRMSM

For $1\leq i\leq m$, $1\leq j\leq n$, \begin{align*} \matrixentry{\conjugate{\alpha A}}{ij} &=\conjugate{\matrixentry{\alpha A}{ij}}&&\knowl{./knowls/definition.CCM.knowl}{\text{Definition CCM}}\\ &=\conjugate{\alpha\matrixentry{A}{ij}}&&\knowl{./knowls/definition.MSM.knowl}{\text{Definition MSM}}\\ &=\conjugate{\alpha}\conjugate{\matrixentry{A}{ij}}&&\knowl{./knowls/theorem.CCRM.knowl}{\text{Theorem CCRM}}\\ &=\conjugate{\alpha}\matrixentry{\conjugate{A}}{ij}&&\knowl{./knowls/definition.CCM.knowl}{\text{Definition CCM}}\\ &=\matrixentry{\conjugate{\alpha}\conjugate{A}}{ij}&&\knowl{./knowls/definition.MSM.knowl}{\text{Definition MSM}} \end{align*}

Since the matrices $\conjugate{\alpha A}$ and $\conjugate{\alpha}\conjugate{A}$ are equal in each entry, Definition ME says that $\conjugate{\alpha A}=\conjugate{\alpha}\conjugate{A}$.