Matrices $A,B$ and $C$ are all $2 \times 2$ matrices and $C=A-CB$. Assuming that $(I+B)^{-1}$ exists, prove that $C=A(I+B)^{-1}$, where $I$ is the $2 \times 2$ identity matrix.
I was wondering if someone could check my work: \begin{align} C=A-CB &\iff C+CB=A \\ &\iff C(I+B)=A \\ &\iff C=A(I+B)^{-1}\\ \end{align}
