Let M be an n⨉n matrix over the reals. Prove that \(\operatorname{rank} M=\operatorname{rank} M^2\) if and only if \(\lim_{\lambda\to 0}  (M+\lambda I)^{-1}M\) exists.

The best solution was submitted by Myeongjae Lee (이명재), 2012학번. Congratulations!

Here is his Solution of Problem 2012-9.

Alternative solutions were submitted by 서기원 (수리과학과 2009학번, +3), 박민재 (2011학번, +3).