J.-H. Wang and P. Y. Wu [Studia Math. 99, No. 2, 115-127 (1991; Zbl 0745.47006)] characterized matrices which are sums of two square-zero matrices, and proved that every matrix with trace-zero is a sum of four square-zero matrices. Moreover, they gave necessary or sufficient conditions for a matrix to be a sum of three square-zero matrices. In particular, they proved that if \(n{\times}n\) matrix \(A\) is a sum of three square-zero matrices, then \(\dim\ker(A-{\alpha}I) \leq 3n/4\) for any scalar \(\alpha \neq 0\).
In the present paper the author shows that the above conditions are not necessarily sufficient for the matrix \(A\) to be a sum of three square-zero matrices, and characterizes the sums of three square-zero matrices among matrices with minimal polynomial of degree \(2\).