Why is the adjugate matrix the null matrix?

Question

I struggle to understand why if $A\in M_n(\mathbb{C}) $, $\det A=0$ and $rank A \le n-2$, then $A^*=O_n$. Could you please tell me why this claim holds? My textbook offers no proof for this.

Answer 1

If $A$ has rank${}<n-1$ then all $(n-1)\times(n-1)$ minors are equal to zero, and so ${\mathrm adj}(A)$ has rank$~0$.