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).
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오타가 있네요. col B = R^k 입니다.