Past exam of the mathematical methods I course of the University of Oxford / 2015 / 12 / iii / Solution

Each Pauli matrix is Hermitian and the coefficients ${\alpha }_j$ are real, so $T^{†}=T$. Taking the conjugate transpose of the absolutely convergent matrix exponential series yieldsThe exponents $-iT/2$ and $iT/2$ commute. Therefore $U^{†}U=e^{-iT/2+iT/2}=I$, proving that $U$ is unitary.