If $A$ is a $(2n+1)\times(2n+1)$ matrix, all of its entries integer, whose diagonal elements are even, and all the other elements odd, is the minimum of the rank of $A$ equal to $2n$ ?
And I'm not sure whether sending to $mod 2$ preserves the rank
2026-03-29 15:31:43.1774798303
rank of an even-diagonal integer matrix
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Hint. It suffices to prove that the leading $2n\times2n$ principal minor is nonzero, but this is evident in modulo 2 arithmetic.