Time

For a spin 1 system l = 1 and m = -1 , 0 , 1. For the operator $S_{z}$ we have

$S_{z}|l,m\rangle$ \Rightarrow Failed to parse (syntax error): {\displaystyle \left \langle l,n| S_z|l,m \rangle = m\hbar \langle l,n|l,m \rangle} $\Rightarrow (S_{z})_{nm}=m\hbar \delta _{nm}$

So

$S_{z}=\hbar {\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\\\end{pmatrix}}$ \Rightarrow $S_{z}^{2}={\hbar }^{2}{\begin{pmatrix}1&0&0\\0&0&0\\0&0&1\\\end{pmatrix}}$