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 \(A\).)

The best solution was submitted by Kim, Kee Taek (김기택, 2015학번). Congratulations!

Here is his solution of problem 2015-12.

Alternative solutions were submitted by 고경훈 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 박지민 (전산학과 대학원생, +3), 엄태현 (수리과학과 2012학번, +3), 이수철 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 진우영 (수리과학과 2012학번, +3), 함도규 (2015학번, +3).