# Let $P ∈ R^{n×m}, Q ∈ R^{m×n}$ . How to show that if I + P Q is nonsingular then I + QP is also nonsingular?

Let $P ∈ \text{Mat}_{n\times m}(\Bbb{R}), Q ∈ \text{Mat}_{m\times n}(\Bbb{R})$ .
Can someone explain to me how to show that if $I + P Q$ is nonsingular then $I + QP$ is also nonsingular?

Nonsingularity means that if $PQ$ exists, then $QP$ is equivalent to that.. So how do I formulate this neatly? It seems pretty obvious to me?

The matrix $I+PQ$ is nonsingular if and only if $-1$ is not an eigenvalue of $PQ$.
Now prove that $PQ$ and $QP$ have the same nonzero eigenvalues.