suppose that A is the matrix in question. we have A = (1/N)B, where B is a matrix with all 1's.
so det(A-xI) = det((1/N)B - xI) = det((1/N)(B - xNI) = det((1/N)(B - NxI)) = ((1/N)^N)(det(B - NxI)).
if we let y = Nx, we have (1/N)^N(det(B - yI)).
so let's look at that determinant, det(B-yI) =
if we subtract the first row from every other row (which does not change the determinant), we get:
if we add every column to the first (which also does not change the determinant), we get:
can you find the determinant of this last matrix? this should give you the only possible eigenvalues rather easily.