I'm stuck on an exercise I got in my linear algebra class where you have to find a $5\times5$ matrix $A$ such that $A$ multiplied by itself five times is the identity matrix $I_5.$ $A$ must not be the identity matrix itself.
My problem is that I don't know how to go about this in a clever way. I know that it makes sense to put only ones on the main diagonal of A and only zeroes below the ones but I don't know how to construct the rest of the matrix. It seems tedious to just try out random examples and I want to do it systematically. Any ideas would be appreciated!