Sum of two strictly n-zero matrices

In this work we investigate when an $n\times n$ Jordan matrix can be written as the sum of two strictly $n$-zero matrices. In particular, we show that if $\mathbb{F}$ is an algebraically closed field of characteristic zero and $A$ is an $n\times n$ matrix over $\mathbb{F}$ with $\text{tr}(A)=0$, then $A$ is the sum of two strictly $n$-zero matrices.