Binary matrix combinatorics

Hello,

the problem is to count how many binary matrices there are with following properties

i. every column must have even number of 's

ii. every row must have even number of 's

For matrices there is none. For matrices there is one:

I think for 3x3 there isn't any. Atleast I couldn't construct one, but I suck at Sudoku's.

The number of square binary matrices is calculated with . So, I though maybe substracting the matrices without even number of 's from that number would help, but then we come the problem, how to pick out the correct matrices.

Any help is appreciated. Thank you!