first of all you should've mentioned that the entries of come from real numbers. to prove that is invertible, we need to prove that if for some vector then

this is easy to prove: thus and hence similarly we see that is invertible too.

let we must show that well, we have and thus henceAnd show that (K-I)(K+I)^-1 is orthogonal (is it's own inverse)

therefore

and has determinant 1

this is false! for example if is a matrix, then and thus in general we can only say that which is obvious because is orthogonal.