To test if a 5x5 Lights Out pattern is solvable, you need to check two things. First look at the top, middle and bottom rows, at the first two and last two lights - the 12 lights bolded here:
[1, 0, 0, 1, 0] [1, 0, ., 1, 0] [1, 1, 1, 1, 0] [., ., ., ., .] [1, 1, 1, 0, 0] [1, 1, ., 0, 0] [0, 0, 1, 0, 0] [., ., ., ., .] [1, 1, 1, 1, 1] [1, 1, ., 1, 1]
If there are an odd number of lights lit up, then the puzzle is unsolvable. In this case 8 of them are lit, which is even, so that does not prove anything yet.
The second test is the same but looking at the first, middle and last columns instead, again only the first two and last two lights:
[1, 0, 0, 1, 0] [1, ., 0, ., 0] [1, 1, 1, 1, 0] [1, ., 1, ., 0] [1, 1, 1, 0, 0] [., ., ., ., .] [0, 0, 1, 0, 0] [0, ., 1, ., 0] [1, 1, 1, 1, 1] [1, ., 1, ., 1]
This time 7 of them are lit up. This is odd, proving this pattern to be unsolvable.
If a pattern passes both tests (i.e. both totals are even numbers) then the pattern can be solved.
To read more about why this works, you can look at my Mathematics of Lights Out page.