Is this correct?
If X and Y are matrices and I is the identity matrix, then
$$I+XY=X^{-1}(I+XY)X$$ $$=X^{-1}IX +X^{-1}XYX=II +IYX$$ $$=I+YX.$$
I am not sure, but I believe there must be a mistake in the first line. Can I assert that?
EDIT: This is part of a problem given to me, does this new information change anything? If $I+XY$ is invertible, show that $I+YX$ is also invertible.
The first line is not true.
We do not have $A=X^{-1}AX$ in general even if $X$ is invertible.
For example let $X = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ and $Y = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}$
$XY=\begin{bmatrix} 0 & 0 \\ -1 & 0 \end{bmatrix}$.
$$I+XY=\begin{bmatrix} 1 & 0 \\ -1 & 1 \end{bmatrix}$$
\begin{align}X^{-1}(I+XY)X&=\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ -1 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \\ &=\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ -1 & -1 \end{bmatrix}\\ &=\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}\\&\neq I+XY\end{align}
Actually, the first line is true if and only if $XY=YX$, assuming that $X$ is invertible. This is what you have proved.
