Spin Operator algebra I am trying to teach myself some QM.

In Christopher J. Cramers textbook Essentials of Computational Chemistry: theory and models, in Appendix C, he goes over Spin algebra.

I am unable to calculate myself that $S^2\alpha = \frac{1}{2}\left( \frac{1}{2}+1 \right) \hbar^2 \alpha$ as well as that $S^2\beta = \frac{1}{2}\left( \frac{1}{2}+1 \right) \hbar^2 \beta$

Where $S^2$ is the spin squared operator and $S = S_x + S_y + S_z$. 

This seems like it should be trivial. It certainly is to him, he says that  repeated application of the following equations lead to the above

$S_x\alpha = \frac{1}{2}\hbar\beta$ 

$S_x\beta= \frac{1}{2}\hbar\alpha $

$S_y\alpha = \frac{1}{2}i\hbar\beta$

$S_y\beta = -\frac{1}{2}i\hbar\alpha$

$S_z\alpha = \frac{1}{2}\hbar\alpha$

$S_z\beta = -\frac{1}{2}\hbar\beta$

my understanding is that $S^2=S_x^2 + S_y^2 + S_z^2$. However, when using this operator on $\alpha$ or $\beta$ I am not sure what the eigenvalues would be..., relatedly, I am not sure even how to operate just $S_x^2$ on $\alpha$.
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}