Spin Operator algebra
One possible way:

As you say
$${\bf{S}^2} = S_x^2+S_y^2+S_z^2$$
Thus
\begin{align}
{\bf{S}^2} &amp;= \left( \frac{\hbar}{2}\right)^2
\left[\begin{pmatrix}
0  &amp; 1 \\
1  &amp; 0 \\
\end{pmatrix}\begin{pmatrix}
0  &amp; 1 \\
1  &amp; 0 \\
\end{pmatrix}
+ 
\begin{pmatrix}
0  &amp; -i \\
i  &amp; 0 \\
\end{pmatrix}
\begin{pmatrix}
0  &amp; -i \\
i  &amp; 0 \\
\end{pmatrix}
+
\begin{pmatrix}
1  &amp; 0 \\
0  &amp; -1 \\
\end{pmatrix}
\begin{pmatrix}
1  &amp; 0 \\
0  &amp; -1 \\
\end{pmatrix}
\right] \\
&amp;=
\left( \frac{\hbar}{2}\right)^2\left[\begin{pmatrix}
1  &amp; 0 \\
0  &amp; 1 \\
\end{pmatrix}+\begin{pmatrix}
1  &amp; 0 \\
0  &amp; 1 \\
\end{pmatrix}+\begin{pmatrix}
1  &amp; 0 \\
0  &amp; 1 \\
\end{pmatrix} \right]\\
&amp;= \left(\frac{3}{4}\right)\hbar^2
\end{align}