Suppose $A \in M_{3\times3}(\mathbb C)$ and $f_A(x) = m_A(x)$ where $f_A(x)$ is the characteristic polynomial of $A$ and $m_A(x)$ is the minimal polynomial of $A$.
If we were to assume $(A+I)^3=0$, would we get a contradiction to the assumption $f_A(x) = m_A(x)$ ?
If not, please give an example of such a matrix.
The polynomial of degree $3,$ $P(x)=(x+1)^3$ annihilates $A$ then since $\mu_A$ is also of degree $3$ then we see that $\mu_A=P=f_A$. An example of such matrix is
$$\begin{pmatrix}-1&1&0\\0&-1&1\\0&0&-1\end{pmatrix}$$
Assuming minimal and characteristic polynomial are the same is never in contradiction with the assumption that the characteristic polynomial is some given polynomial$~P$, provided $P$ is possible at all (so $P$ must be monic and of the correct degree). With the combined assumptions $P$ will of course also be the minimal polynomial. In this concrete case $P=(X+1)^3=X^3+3X^2+3X+1$ may very well be the minimal (and characteristic) polynomial of a $3\times3$ matrix$~A$, so that $(A+1)^3=0$. The standard example that this is possible is the companion matrix for $P$, which in this case is $$A=\begin{pmatrix}0&0&-1\\1&0&-3\\0&1&-3\end{pmatrix},$$ since the minimal and characteristic polynomial of this companion matrix are both equal to$~P$.
