For $=1,$ the angular$-$momentum components L$$ x$,$ L$$ y and L$$ z are represented by the matrices Lx $=0$ ~$/20$ ~$/20$ ~$/20$ ~$/20$ Ly $=0$i~$/20$ i~$/20$i~$/20$ i~$/20$ Lz $=$ ~ $0000000$~ $,($i$)$ Compute L $2=$ L $2$ x $+$ L $2$ y $+$ L $2$ z and show that it is a multiple of the identity matrix. $($ii$)$ Compute the eigenvalues of Lx and Ly$.($iii$)$ Show that $101$ is an eigenvector of Ly$.$ If a system is in this state, what are the probabilities for finding the values ~$,0,$ and $$~$,$ respectively, in a single measurement of Lz$?$