\(
cov(X) = \mathbf{E}[(X-\mathbf{E}(X))(X-\mathbf{E}(X))^T] \\
\)
so,
\(
\begin{equation}
\begin{split}
cov(AX) & = \mathbf{E}[(AX-\mathbf{E}(AX)(AX-\mathbf{E}(AX))^T] \\
& = \mathbf{E}[A(X-\mathbf{E}(X))(X-\mathbf{E}(X))^T A^T] \\
& = A\mathbf{E}[(X-\mathbf{E}(X))(X-\mathbf{E}(X))^T]A^T \\
& = Acov(X)A^T
\end{split}
\end{equation}
\)