Commutation Relations for Spin Operators

A commutator of two observables is defined as:

\[[\hat{A},\hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}\]

Determine the results of the following commutators:

1. \([\hat{S}_x,\hat{S}_y]\)
   
2. \([\hat{S}_y,\hat{S}_z]\)
   
3. \([\hat{S}_z,\hat{S}_x]\)
   
4. \([\hat{S}_y,\hat{S}_x]\)
   
5. \([\hat{S}_z,\hat{S}_y]\)
   
6. \([\hat{S}_x,\hat{S}_z]\)
   

Remember that the matrix representation of the spin operators written in the \(S_z\) basis is: \begin{align*} \hat{S}_x \doteq \frac{\hbar}{2}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \quad \hat{S}_y \doteq \frac{\hbar}{2}\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} \quad \hat{S}_z \doteq \frac{\hbar}{2}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \end{align*}