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    $\begingroup$ We may finish the proof after your second matrix, since the determinant of a block-triangular matrix is a product of determinants of diagonal blocks. $\endgroup$ Commented May 21, 2016 at 18:34
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    $\begingroup$ @FedorPetrov I only remembered that after typing my answer and was wondering whether to edit it or not. I will edit my answer now, in order to shorten it and save readers' time. $\endgroup$ Commented May 21, 2016 at 18:42