Question

If A is an 7 x 7 matrix with all 49 entries being odd numbers. Show that det (A) is a multiple of 64. You may use the fact (without proving) that an n x n matrix with integer entries has an integer determinant.

Answer 1

First we subtract first rowfrom other rows. Thus we obtain six rows with all even entries.
If we divide each of these rows by 2 then we have to multiply det(A) by 2^6=64.

Now det(A)=64*det(B), where all entries of B are again integer, so det(B) is
integer too.
And then detA is integer and multiply of 64.