# 2015-12 Rank

Let $$A$$ be an $$n\times n$$ matrix with complex entries. Prove that if $$A^2=A^*$$, then $\operatorname{rank}(A+A^*)=\operatorname{rank}(A).$ (Here, $$A^*$$ is the conjugate transpose of $$A$$.)

(This is the last problem of this semester. Thank you for participating KAIST Math Problem of the Week.)

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