Prove that, for any $\theta$, $\lambda$, $\mu$
$Q = \pmatrix{\cos{\lambda}\cos{\mu} - \cos{\theta}\sin{\lambda}\sin{\mu}&\sin{\lambda}\cos{\mu} + \cos{\theta}\cos{\lambda}\sin{\mu}&\sin{\theta}\sin{\mu}\\-\cos{\lambda}\sin{\mu} - \cos{\theta}\sin{\lambda}\cos{\mu}&-\sin{\lambda}\sin{\mu} + \cos{\theta}\cos{\lambda}\cos{\mu}&\sin{\theta}\cos{\mu}\\ \sin{\theta}\sin{\lambda}&-\sin{\theta}\cos{\lambda}&\cos{\theta}}$
is a proper orthogonal matrix and write down a formula for $Q^{-1}$
How will I be able to do this problem? I know that in order to be a proper orthogonal matrix it must have $\det(Q) = 1$ and it must form orthonormal columns, but how can I show that here?