Let $A$ be a matrix belonging to $\mathbf{GL}_n$. Let $x$ be a nonzero real number. Express $\det(x I_n - A)$ in terms of $x$, $\det A$ and $\det\left(\frac{1}{x} I_n - A^{-1}\right)$.
Let $A$ be a matrix belonging to $\mathbf{GL}_n$. Let $x$ be a nonzero real number. Express $\det(x I_n - A)$ in terms of $x$, $\det A$ and $\det\left(\frac{1}{x} I_n - A^{-1}\right)$.