# Number of ways of filling $n \times n$ binary matrix

In how many ways can we fill $n \times n$ matrix with $+1$ or $-1$ such that the product of the entries in each row and each column equals $-1$?

• I have tried upto : In 2*2 matrix No. of ways of filling such arrangement is 2; In 3*3 matrix No. of ways of filling such arrangement is 16; but i have done it by trial and error (i.e, no valid process); so how can i solve this problem – Suresh Jan 22 '18 at 14:20
• Here is a related question. – Rodrigo de Azevedo Apr 21 '18 at 18:44

Let the matrix size be $n$.
For each row and each column, there must be an odd number of $(-1)$s. (Why?)
You can arbitrarily assign $1$ and $-1$ for each element of the upper-left $(n-1)\times(n-1)$ cells. The last column will have to balance its previous rows (aka make the row product $-1$), and the last row will have to balance the previous columns (aka make the column product $-1$).