## 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*)