Can a non-zero matrix have a zero characteristic polynomial?

Question

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 $5$.

Answer 1

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).

Answer 2

Polynomials are not functions. $t^5+4t$ is not the zero polynomial over $\Bbb Z_5$. It evaluates to zero everywhere, so it becomes the zero function. But it is not the zero polynomial, as it has non-zero coefficients.

Thus, the argument you have already come across proves that the characteristic polynomial of a matrix can never be the zero polynomial.