Decomposition of matrices into invertible and square-zero matrices

In this paper we show that any matrix $A$ in $\mathbb{M}_n(\mathbb{F})$ over an arbitrary field $\mathbb{F}$ can be decomposed as a sum of an invertible matrix and a nilpotent matrix of order at most two if and only if its rank is at least $\frac n2$. We also study when $A$ can be decomposed as the sum of a torsion matrix and a nilpotent matrix of order at most two. For fields of prime characteristic, we show that this second decomposition holds as soon as the characteristic polynomial of $A$ is algebraic over its base field and the rank condition is fulfilled, and we present several examples that show that the decomposition does not hold in general. Furthermore, we completely solve the second decomposition problem for nilpotent matrices over arbitrary fields. This somewhat continues our recent publications in Lin. Multilin. Algebra (2023) and Internat. J. Algebra Computat. (2022) as well as it strengthens results due to Calugareanu-Lam in J. Algebra Appl. (2016).