The unitary property of the Pauli matrices

Question Q6.5.1

Demonstrate that the three Pauli matrices given in below are unitary. That is, that $\sigma_p^\dagger\sigma_p = I_2$ for $p = x,y,z$, where $I_2$ is the $2\times 2$ identity matrix and $\dagger$ denotes the Hermitian conjugate (conjugate transpose).

\begin{align*} \sigma_x = \left( \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma_y = \left( \begin{array}{rr} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma_z = \left( \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right). \end{align*}


Solution