Lights Off, Revisited!

Johnny is, by now, a little bored of the original Lights Off game, and he has proposed his own variant. This time, the task which awaits him is much, much harder... Imagine we have a large square board, containing an n x n grid (matrix) of lightbulbs. Initially, some of them are on, and the rest are off. We can perform some special operations on the whole matrix, and we would like to switch off a