Let $\mathbb{P}_3$ be the vector space of polynomials of degree at most 3 in one variable. For $p \in\mathbb{P}_3$ define $T(p)=p^{''}$ (the 2nd derivative).
I want to find the characteristic polynomial and the minimal polynomial of this operator. I found the matrix of $T$ with respect to the basis $\left\{1, x, x^2, x^3\right\}$ and obtained:
$M=\left( \begin{array}{ccc} 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 6 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right) $
I found the characteristic polynomial of this matrix by finding $\det (M-\lambda I)$. I obtained $\text{char(T)}=\lambda^4$.
Now I now that the characteristic polynomial is divisible by the minimal polynomial and that the minimal polynomial is the monic polynomial, lets call it $P$, of least degree such that $P(M)=0$. The only divisors of $\text{char(T)}=\lambda^4$ are $\lambda^4$, $\lambda^3$, $\lambda^2$, $\lambda$, and 1. How would any of these map $M$ to zero? Thank you!!