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.
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2012-9 Rank of a matrix,
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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.