Given the matrices for C₃ and σ_h below,

${\mathrm{C}}_3=\left[\begin{array}{c}-1/2\\ \sqrt{3}/2\\ 0\end{array}\begin{array}{c}-\sqrt{3}/2\\ -1/2\\ 0\end{array}\begin{array}{c}0\\ 0\\ 1\end{array}\right]{\sigma }_{\mathrm{h}}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 2& 0& -1\end{array}\right]$

the trace of the matrix representing for $S_3^2$ is

Concept:-

• An action that leaves an object looking the same after it has been carried out is called a symmetry operation.
• The symmetry operations include rotation, reflection, the alternating axis of symmetry, and inversions.
• A molecule possessing an alternative axis of symmetry (S_n) of n fold represents the rotation of the molecule about the axis $\frac{{360}^{\circ }}{n}$ followed by reflection through a plane perpendicular to this axis produces an indistinguishable structure.
• S_n It can be represented by,

$S_n=C_n\times \sigma$ 

• The trace of a matrix A, designated by tr(A), is the sum of the elements on the main diagonal.
• If matrix A is represented by the matrix,

$A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 2& 0& -1\end{array}\right]$

then the trace of matrix A is given by,

tr(A)= 1+1-1

=1

Explanation:-

• The alternative axis of symmetry (Sn) is represented by,

$S_n=C_n\times \sigma$​

• Now, $S_3^2$ is given by,

​$S_n^2=C_n^2\times {\sigma }^2$

$=C_n^2\times E\left(As,{\sigma }^2=E\right)$

$=C_n^2$

• Thus, trace of the matrix $S_3^2$ is given by,
   

${\mathrm{S}}_3^2={\mathrm{C}}_3^2{\sigma }_{\mathrm{n}}^2={\mathrm{C}}_3^2$

${\mathrm{C}}_3=\left[\begin{array}{c}-1/2\\ \sqrt{3}/2\\ 0\end{array}\begin{array}{c}-\sqrt{3}/2\\ -1/2\\ 0\end{array}\begin{array}{c}0\\ 0\\ 1\end{array}\right]$

${\mathrm{C}}_3^2=\left[\begin{array}{c}-1/2\\ \sqrt{3}/2\\ 0\end{array}\begin{array}{c}-\sqrt{3}/2\\ -1/2\\ 0\end{array}\begin{array}{c}0\\ 0\\ 1\end{array}\right]\left[\begin{array}{c}-1/2\\ \sqrt{3}/2\\ 0\end{array}\begin{array}{c}-\sqrt{3}/2\\ -1/2\\ 0\end{array}\begin{array}{c}0\\ 0\\ 1\end{array}\right]$

$=\left[\begin{array}{ccc}\frac{1}{4}-\frac{3}{4}+0& \frac{\sqrt{3}}{4}+\frac{\sqrt{3}}{4}+0& 0+0+0\\ -\frac{\sqrt{3}}{4}-\frac{\sqrt{3}}{4}+0& -\frac{3}{4}+\frac{1}{4}+0& 0+0+0\\ 0+0+0& 0+0+0& 0+0+0\end{array}\right]$

$=\left[\begin{array}{ccc}-\frac{1}{2}& \frac{\sqrt{3}}{2}& 0\\ -\frac{\sqrt{3}}{2}& -\frac{1}{2}& 0\\ 0& 0& 1\end{array}\right]$

Trace of matrix $S_3^2$ is,

$-\frac{1}{2}-\frac{1}{2}+1$

$=0$

Conclusion:-

Hence, trace of the matrix representing for $S_3^2$ is 0