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let $A=[a_{ij}]\in M_n(\mathbb R)$ such that $a_{ij}\in\{-1,1\} $ then how prove $$2^{n-1}|\ \det(A)$$

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up vote 2 down vote accepted

Hint: Use elementary row operations to kill all but the first entries in the first column. Then you get an $(n-1)\times(n-1)$ submatrix with each entry divisible by $2$.

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