Let $ A \in M_{n}(\mathbb{C}) $ so that $ rank(4I_{n} + 5A) \leq rank(2I_{n} + 3A) $.
Prove that $ rank(2I_{n} + 3A) \geq 1 + [\frac{n}{2}]$.
2026-04-02 06:18:14.1775110694
Rank of matrix inequality
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This is not correct. Consider $$ A = \begin{pmatrix}-\frac 4 5 & 0\\0 & -\frac 2 3\end{pmatrix}. $$