Eigenvalues of inverse matrix to a given matrix

How to calculate the eigenvalue of the inverse of a matrix given matrix is $A= \begin{bmatrix} 0&1&0\\ 0&0&1 \\4&-17&8\end{bmatrix}$

Is there any fast method?

$$ A^{-1} x = \lambda x \iff \\ x = \lambda A x \Rightarrow \\ A x = (1/\lambda) x $$ So the non-zero eigenvalues of $A^{-1}$ and $A$ are related, are the multiplicative inverse to each other.

Looking for the eigenvalues of $A$ we get the characteristic polynomial $$ (-\lambda)((-\lambda)(8-\lambda)+17) - 4 = \\ (-\lambda)(\lambda^2 -8\lambda +17) + 4 = \\ -\lambda^3 + 8\lambda^2-17\lambda + 4 $$ Getting the roots by guessing, using the complicated formulas for cubic equations or some numerical procedure or using a computer algebra system gives $\lambda \in \{ 4, 2\pm\sqrt{3} \}$.

The eigenvalues of $A^{-1}$ are inverse, thus $\{ 1/4, 1/(2\pm\sqrt{3})\}$.

Remark: If $A$ is invertible it can not have a zero eigenvalue, because $A x = 0 x = 0$ for some eigenvector $x \ne 0$ would mean its kernel would have a dimension larger than zero and $A$ thus would not have full rank, contradicting it being invertible.