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A note on sums of three square-zero matrices

It is known that every complex trace-zero matrix is the sum of four square-zero matrices, but not necessarily of three such matrices. In this note, we prove that for every trace-zero matrix $A$ over an arbitrary field, there is a non-negative integer $p$ such that the extended matrix $A \oplus 0_p$ is the sum of three square-zero matrices (more precisely, one can simply take $p$ as the number of rows of $A$). Moreover, we demonstrate that if the underlying field has characteristic $2$ then every trace-zero matrix is the sum of three square-zero matrices. We also discuss a counterpart of the latter result for sums of idempotents.