$\begin{align*} 
(E'A')^{T} &= ((M^{T}M)(M^{-1}AM))^{T} = \\
&= (M^{T}AM)^{T} = \\
&= M^{T}A^{T}M^{TT} = \\
&= M^{T}AM = \\
&= M^{T}M(M^{-1}AM) = \\
&= E'A'
\end{align*}$