Compute $[L_x,L_y]$ Using Matrices.

$$\bgroup\color{black}[L_x,L_y]={\hbar^2\over 2i} \left( \left... ...t) \left(\matrix{0&1&0\cr 1&0&1\cr 0&1&0}\right) \right)\egroup$$

$$\bgroup\color{black}{\hbar^2\over 2i} \left( \left(\matrix{-... ...left(\matrix{1&0&0\cr 0&0&0\cr 0&0&-1}\right)=i\hbar L_z\egroup$$

The other relations will prove to be correct too, as they must. Its a reassuring check and a calculational example.