Prove that if $\mathrm{rank}(A-I_n)=p$ and $\mathrm{rank}(B-I_n)=q$, then $\mathrm{rank}(AB-I_n)\leq p+q$

Question

Let $A,B\in M_n\mathbb{(R)}$ be matrices with $\mathrm{rank}(A-I_n)=p$ and $\mathrm{rank}(B-I_n)=q$. Show that $\mathrm{rank}(AB-I_n)\leq p+q$.

I have no idea how to relate the two rank given to solve the question stated.

Answer 1

We have $\DeclareMathOperator{\rank}{rank}$ $$ \rank(AB - I) =\\ \rank(AB - A + A - I)=\\ \rank(A(B-I) + A-I) \leq\\ \min\{\rank(A),\rank(B-I)\} + \rank(A-I) \leq \\ \rank(B-I) + \rank(A-I) =\\ p+q $$