It's true when $A$ is a matrix over $\mathbb{R}$, see Prove rank $A^TA$ = rank $A$ for any $A_{m \times n}$
2026-03-27 07:16:59.1774595819
Does $\operatorname{rank} {A^T}A = \operatorname{rank} A$ holds when $A$ is matrix over finite field?
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How about $A=\pmatrix{1&1\cr1&1}$ over $\Bbb F_2$?