How do I rearrange this matrix equation to find X?

Question

Given that the matrices $D$, $E$ and $F$ are invertible, how do I rearrange the equation to solve for $X$ when $D(X+3I)E = 5D(F+E) +E^2$.

Would I just take the inverse of $D$ and $E$ to both sides leaving me with $X= 5D(F+E)D^{-1}E^{-1} +EI - 3I $?

Answer 1

When you multiply on the left by $D^{-1}$ on the left side, it should be on the left on the right side as well. Then $D^{-1}5D=5$ and there shouldn't be any $D$s in the answer. You have commuted the $(F+E)$ and the $D^{-1}$