# 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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# 2015-4 An inequality on positive semidefinite matrices

Let $$M=\begin{pmatrix} A & B \\ B^*& C \end{pmatrix}$$ be a positive semidefinite Hermian matrix. Prove that $\operatorname{rank} M \le \operatorname{rank} A +\operatorname{rank} C.$ (Here, $$A$$, $$B$$, $$C$$ are matrices.)

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# 2014-18 Rank

Let $$A$$ and $$B$$ be $$n\times n$$ real matrices for an odd integer $$n$$. Prove that if both $$A+A^T$$ and $$B+B^T$$ are invertible, then $$AB\neq 0$$.

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# 2012-9 Rank of a matrix

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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