  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.
