The Pauli spin matrices in quantum mechanics are given by the following matrices: (sigma_{1}=left(begin{array}{ll}0 & 1 1 & 0end{array} ight), sigma_{2}=left(begin{array}{cc}0 & -i

The Pauli spin matrices in quantum mechanics are given by the following matrices: \(\sigma_{1}=\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right), \sigma_{2}=\left(\begin{array}{cc}0 & -i \\ i & 0\end{array}\right)\), and \(\sigma_{3}=\left(\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right)\). Show

a. \(\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma_{3}^{2}=I\).

b. \(\left\{\sigma_{i}, \sigma_{j}\right\} \equiv \sigma_{i} \sigma_{j}+\sigma_{j} \sigma_{i}=2 \delta_{i j} I\), for \(i, j=1,2,3\) and \(I\) the \(2 \times 2\) identity matrix. \(\{\),\(\} is the anti-commutation operation.\)

c. \(\left[\sigma_{1}, \sigma_{2}\right] \equiv \sigma_{1} \sigma_{2}-\sigma_{2} \sigma_{1}=2 i \sigma_{3}\), and similarly for the other pairs. [,] is the commutation operation.

d. Show that an arbitrary \(2 \times 2\) matrix \(M\) can be written as a linear combination of Pauli matrices, \(M=a_{0} I+\sum_{j=1}^{3} a_{j} \sigma_{j}\), where the \(a_{j}\) 's are complex numbers.