Jordan Canonical Form of $A^{-1}$ 
Let $A$ be a $5 \times 5$ invertible matrix over $\mathbb{R}$. Suppose
  $c_A(x)=(x+1)^3(x+2)^2$ and $m_A(x)=(x+1)(x+2)^2$, what is a Jordan
  Canonical Form of $A^{-1}$?

Is it possible to obtain a JCF of $A^{-1}$ from that of $A$, or we have to explicitly solve for $c_{A^{-1}}(x)$ and $m_{A^{-1}}(x)$ in the first place? 

Its easy to see that 
$$A=\begin{pmatrix}
-2&1&0&0&0\\
0&-2&0&0&0\\
0&0&-1&0&0\\
0&0&0&-1&0\\
0&0&0&0&-1\\
\end{pmatrix}$$
so we see
$$A^{-1}=\begin{pmatrix}
-\frac{1}{2}&-\frac{1}{4}&0&0&0\\
0&-\frac{1}{2}&0&0&0\\
0&0&-1&0&0\\
0&0&0&-1&0\\
0&0&0&0&-1\\
\end{pmatrix}$$
