# 2020-01 Another singular matrix

For a given positive integer $$n$$, find all non-negative integers $$r$$ such that the following statement holds:

For any real $$n \times n$$ matrix $$A$$ with rank $$r$$, there exists a real $$n \times n$$ matrix $$B$$ such that $$\det (AB+BA) \neq 0$$.

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Let $$A, B$$ be $$n \times n$$ Hermitian matrices. Find all positive integer $$n$$ such that the following statement holds:
“If $$AB – BA$$ is singular, then $$A$$ and $$B$$ have a common eigenvector.”