Inequality Regarding Ranks of Matrices
Problem 58
Let $A$ be an $n \times n$ matrix over a field $K$. Prove that
\[\rk(A^2)-\rk(A^3)\leq \rk(A)-\rk(A^2),\]
where $\rk(B)$ denotes the rank of a matrix $B$.
(University of California, Berkeley, Qualifying Exam)
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