Question

If X and Y are n*n matrices, Show that (I-XY) is invertible if and only if (I-YX) is
invertible.
Hint: Use X(I-YX)=(I-XY)X

Answer 1

Let we have that $I - XY \in GL_n(R)$ . Then

$$\begin{array}{l} \left(I - Y X\right) \left(I + Y \left(I - X Y\right) ^ {- 1} X\right) = I + Y \left(I - X Y\right) ^ {- 1} X - Y X - Y X Y \left(I - X Y\right) ^ {- 1} X = \\ = I + Y (I - X Y) (I - X Y) ^ {- 1} X - Y X = I \\ \end{array}$$

So, $(I - YX)$ is invertible. Replacing $X$ and $Y$ in last formula shows reverse conclusion.