Prove if $\operatorname{adj}(A) =0$, then $\operatorname{rank}A \leq n-2$

My initial idea is to setup an induction. Base case $A \in M\_{2×2}$ which obviously implies that $A = [O]\_{2×2} \rightarrow \operatorname{rank}A \leq 0 = n-2$. However, I can't seem to reason how assuming it worked in the $M\_{n\times n}$ case allows for the $M\_{(n+1)×(n+1)}$ case to work....