If $λ \neq 0$ an eigenvalue of matrix $A$ then find an eigenvalue of $\text{adj}\left(A\right)$

Question

Any ideas? Do we use the definition of the eigenvalues?

Answer 1

Let us call $\text{adj}(A)=B$ and $\text{det}(A)=d$. Then $BA=dI_n$. Suppose $v$ is an eigen vector corresponding to the eigen value $\lambda$ of $A$. Then \begin{align*} BAv & = B(\lambda v)\\ &=\lambda Bv. \end{align*} But \begin{align*} BAv & = d I_n v\\ &=dv. \end{align*} Thus $Bv=\frac{d}{\lambda}v.$ Hence $v$ is an eigen vector of $B$ with the eigen value $\frac{d}{\lambda}$.

Answer 2

Hint: use the equation $$ \text{adj}(A) A = \det(A) I_n $$