Lights Out Variant: Flipping the whole row and column. So I found this puzzle similar to Lights Out, if any of you have ever played that.  Basically the puzzle works in a grid of lights like so:

1 0 0 00 0 0 00 1 0 0 0 0 1 0

When you selected a light (the X), it toggled itself and all the lights in its row and column:

1 0 1 01 1 X 10 1 1 0 0 0 0 0

This got me wondering how one could tell whether there was a solution for a given setup and grid size, and if so, what was it? I can't seem to get anywhere. Could anyone push me in the right direction?

Think of each light as a variable taking values in $0, 1$. Flipping a switch has the effect of adding $1$ to each corresponding light, modulo $2$. To the switch at position $i, j$ make correspond a variable $s_{i,j}$representing the number of times you flip that switch. Now the final value of each light is equal to its initial value plus each $s_{i,j}$ that shares a column or row with it - this is a linear equation in the variables $s_{i,j}$. For an $m\times n$ grid, we thus obtain $mn$ linear equations (one for each light) in $mn$ variables (one for each switch).
The integers modulo $2$ are a field, so all of the usual results of linear algebra should apply.
Example. Let's works out the $2\times2$ case. Which grids are solvable?
If we represent "on" as $1$ and "off" as $0$, then note that we need to add each light to itself. That is, if a light starts at value $0$, we need to flip the switches so as to add $0$ to that light (because we don't want it to change), but if it starts at value $1$, we need to flip the switches so as to add $1$ to that light. So the equation for each light is going to look like sum of associated switches = initial state.
Writing the initial state of the $i,j$ light as $l_{i,j}$, we therefore need to solve:
$$l_{1,1}=s_{1,1}+s_{1,2}+s_{2,1}$$
$$l_{1,2}=s_{1,2}+s_{1,1}+s_{2,2}$$
$$l_{2,1}=s_{2,1}+s_{1,1}+s_{2,2}$$
$$l_{2,2}=s_{2,2}+s_{1,2}+s_{2,1}$$
The matrix for this system looks like this (omitting zeros for readability):
\begin{array}{cccc}
1 & 1 & 1 & \\
1 & 1 & & 1 \\
1 & & 1 & 1 \\
& 1 & 1 & 1
\end{array}
Gaussian Elimination works easily on this matrix (tip: remember that mod $2$, $a + a = 0$ for any $a$) and we obtain an explicit solution for the general $2\times2$ grid:
$$s_{1,1}=l_{1,1}+l_{1,2}+l_{2,1}$$
$$s_{1,2}=l_{1,2}+l_{1,1}+l_{2,2}$$
$$s_{2,1}=l_{2,1}+l_{1,1}+l_{2,2}$$
$$s_{2,2}=l_{2,2}+l_{1,2}+l_{2,1}$$
So the answer is that all $2\times2$ grids are solvable, and there's the solution (concurring with Aryabhata's proposition).
As the number of equations is equal to the number of cells on the grid, this method quickly becomes too tedious and complicated to perform by hand. I would try to investigate the determinant of these matrices in the abstract case and see if you come up with anything. That would at least tell you which grid sizes have a guaranteed solution.
