# If a $n \times n$ matrix $A$ is not invertible, then is it possible that the classical adjoint matrix of $A$ is invertible?

Let $$A$$ be a $$n \times n$$ matrix over $$\mathbb{R}$$ or $$\mathbb{C}$$.

Let $$\text{adj}(A)$$ denote the classical adjoint matrix of $$A$$ (or adjugate matrix of $$A$$).

It is known that $$A \cdot \text{adj}(A) = \text{det}(A) I_n = \text{adj}(A) \cdot A$$.

Thus, if $$A$$ is invertible, then $$\text{det}(A) \neq 0$$, and so $$\text{adj}(A)$$ is also invertible,

since $$\frac{A}{\text{det}(A)} \cdot \text{adj}(A) = I_n = \text{adj}(A) \cdot \frac{A}{\text{det}(A)}$$.

My question :

Is there a case that $$A$$ is not invertible and $$\text{adj}(A)$$ is invertible?

Suppose $$\operatorname{adj}(A)$$ is invertible but $$A$$ is not. Then $$\det(A)=0$$, and $$A\operatorname{adj}(A)=\det(A)I$$ implies $$A=\det(A)\left(\operatorname{adj}(A)\right)^{-1}$$, i.e., $$A$$ is the zero matrix. In this case it's easy to see $$\operatorname{adj}(A)$$ is also the zero matrix (all of the cofactors of $$A$$ are zero), contradicting the assumption.