Given the matrices for C₃ and σ_h below,

\(\rm C_3 = \left[ {\begin{array}{*{20}{c}} { - 1/2}\\ { \sqrt 3 /2}\\ 0 \end{array}\begin{array}{*{20}{c}} { - \sqrt 3 /2}\\ { - 1/2}\\ 0 \end{array}\begin{array}{*{20}{c}} 0\\ 0\\ 1 \end{array}} \right]\sigma_h = \left[ {\begin{array}{*{20}{c}} 1&0&0\\ 0&1&0\\ 2&0&{ - 1} \end{array}}\right]\)

the trace of the matrix representing for \(S^2_3\) 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 \({{{{360}^ \circ }} \over n}\) 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,

\(A = \left[ {\begin{array}{*{20}{c}} 1&0&0\\ 0&1&0\\ 2&0&{ - 1} \end{array}}\right] \)

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, \(S^2_3\) 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 \(S^2_3\) is given by,
   

\(\rm S_3^2=C_3^2\sigma_n^2=C_3^2\)

\(\rm C_3 = \left[ {\begin{array}{*{20}{c}} { - 1/2}\\ { \sqrt 3 /2}\\ 0 \end{array}\begin{array}{*{20}{c}} { - \sqrt 3 /2}\\ { - 1/2}\\ 0 \end{array}\begin{array}{*{20}{c}} 0\\ 0\\ 1 \end{array}} \right] \)

\(\rm C_3^2 = \left[ {\begin{array}{*{20}{c}} { - 1/2}\\ { \sqrt 3 /2}\\ 0 \end{array}\begin{array}{*{20}{c}} { - \sqrt 3 /2}\\ { - 1/2}\\ 0 \end{array}\begin{array}{*{20}{c}} 0\\ 0\\ 1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} { - 1/2}\\ { \sqrt 3 /2}\\ 0 \end{array}\begin{array}{*{20}{c}} { - \sqrt 3 /2}\\ { - 1/2}\\ 0 \end{array}\begin{array}{*{20}{c}} 0\\ 0\\ 1 \end{array}} \right] \)

\(\rm =\begin{bmatrix}\frac{1}{4}-\frac{3}{4}+0&\frac{\sqrt3}{4}+\frac{\sqrt3}{4}+0&0+0+0\\\ -\frac{\sqrt3}{4}-\frac{\sqrt3}{4}+0&-\frac{3}{4}+\frac{1}{4}+0&0+0+0\\\ 0+0+0&0+0+0&0+0+0\end{bmatrix} \)

\(\rm =\begin{bmatrix}-\frac{1}{2}&\frac{\sqrt3}{2}&0\\\ -\frac{\sqrt3}{2}&-\frac{1}{2}&0\\\ 0&0&1\end{bmatrix} \)

Trace of matrix \(S^2_3\) is,

\( - {1 \over 2} - {1 \over 2} + 1\)

\( = 0\)

Conclusion:-

Hence, trace of the matrix representing for \(S^2_3\) is 0