# how to prove $2^{n-1}|\det(A)$ where $A=[a_{ij}]\in M_n(\mathbb R)$ and $a_{ij}\in\{-1,1\}$

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

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 $2$.