Keep order of mulciplication in Matrix production

{{λ E, B}, {λ A1, λ E}}.{{E, 0}, {-A1, E}}

\begin{align*}\left(\begin{array}{cc} E^2 \lambda -A1 B & B E \\ 0 & E^2 \lambda \\\end{array}\right)\end{align*}

The result is not quit true, since $A1 B$ may not equal to $B A1$ when $A1$ and $B$ are block matrix.