Shifting matrix

An $n \times n$ matrix is filled with numbers, where there are $k$ number ones and the rest zero. The operation that is allowed on the matrix is to shift a single column down by one (so that each number in that column moves down by one, and the bottom most number goes to the top), or to shift a single row to the left by one.

Determine all $k$ such that, no matter how the initial condition is, through a series of operations, we can make it so that each column and each row has an even sum.