Let $A\in M_{n\times n}$ be and invertible matrix over complex field and we assume it's already at Jordan form where $B=\{v_1,…,v_n \}$ is Jordan basis for A.
Find Jordan form and Jordan basis for $A^{-1}$
I think I can show that Jordan form will be almost the same with difference that eigenvalue $\lambda$ will be replaced by $\frac{1}{\lambda}$ at diagonal but I don't have idea how basis for this will looks like.