Prove that $\mathrm{adj}(\mathrm{adj}(A)) = (\mathrm{det}(A))^{n-2} \cdot A$ for $\det A = 0$

Question

Prove that $\mathrm{adj}(\mathrm{adj}(A)) = (\mathrm{det}(A))^{n-2} \cdot A$

Hello, this question exists here.

In the topmost answer, when we reach at $$\det A \cdot I_n \cdot \operatorname{adj}(\operatorname{adj}(A))=(\det A)^{n-1}\cdot A.$$

and if $\det A = 0$, what do we do?

Sorry if this is a stupid question.

Answer 1

Both sides of the identity are matrices whose elements are polynomial in the elements of $A,$ so the result follows by continuity.