How many $n$ x $n$ matrices with this given property? I would like to know how many $n$ x $n$ matrices are there containing elements that are either $1$ or $-1$, such that the product of the elements in each row and column is $-1$?

Hint:  In the first column, you can pick the first $n-1$ elements any way you want, then the last element is fixed.    You need an odd number of $-1$'s in the column.  Similarly for the rest of the first $n-1$ columns.  Then the  top $n-1$ cels of the last column are fixed.  What about the lower right cell?
