Are there some field $\mathbb{F}$, some $n \in \{1,2,\dots\}$, and some non-zero $n \times n$ matrix $A$ over $\mathbb{F}$, whose characteristic polynomial $p_A(t)$ is identically $0$?
The same question was asked here in the past, and the answer explained that such a $p_A(t)$ was impossible, because a characteristic polynomial of an $n\times n$ matrix had degree $n$.
But this answer is unsatisfactory, because in some cases an identically zero polynomial has a positive degree: take for instance the polynomial $p(t) = t^5 + 4t$ in the field $\mathbb{Z}/5\mathbb{Z}$ of the integers modulo $5$.
Best Answer
If you are asking about characteristic polynomials, then the answer to the question that you have mentioned is correct: it has degree $n$, and therefore it cannot be the null polynomial.
But if you are talking about polynomial functions, then, yes, the polynomial function corresponding to the characteristic polynomial of a matrix can by the null function. Take, for instance $A=\left[\begin{smallmatrix}0&0\\0&1\end{smallmatrix}\right]$. Then $p_A(t)=t^2-t$. So, if you are working over the field $\Bbb F_2$, you will get the null function (but not the null polynomial).