Show that $a(XY)=X(aY)$ where $a$ is an element of field and $X$ and $Y$ are $m\times n$ and $n\times p$ matrices Show that $a(XY)=X(aY)$ where $a$ is an element of field and $X$ and $Y$ are $m\times n$ and $n\times p$  matrices.
By using the notation to designate the coefficient of the $i$th row and $j$th column of a matrix $A$: $(A)_{i,j}$, we have

$$(a(XY))_{i,j}=a\cdot (XY)_{i,j}.$$

On the other hand,

\begin{align}
(X(aY))_{i,j} &amp; =\sum_{k=1}^{n} (X)_{i,k} (aY)_{k,j}
\\ &amp; =\sum_{k=1}^{n} (X)_{i,k}\  a\cdot(Y)_{k,j}
\\ &amp; = a\cdot\sum_{k=1}^{n} (X)_{i,k} (Y)_{k,j} = a\cdot(XY)_{i,j}.
\end{align}
Matrix multiplication $\pi:(X,Y)\mapsto XY$ is linear in each slot. By definition $X(aY) = \pi(X,aY)$, which by linearity in the second slot equals $a \pi (X,Y)  = a (X Y)$. Hence $a(X Y) = X(aY)$.