Given the matrices for C₃ and σh below C₃= -1/2 -(√(3))/2 0 (√(3))/2 -1/2 0 0 0 1 σh= 1 0 0 0 1 0 0 0 1 the trace of the matrix representing for S₃² is

Q43 · CSIR-NET Chemistry, December 2019

Paper: CSIR-NET December 2019 · Subject: Physical Chemistry · Chapter: Group Theory · Topic: Group Theory – General · Marks: 2 · Difficulty: Medium

Given the matrices for $C_3$ and $\sigma_h$ below

\[ C_3=\begin{pmatrix} -1/2 & -\dfrac{\sqrt{3}}{2} & 0 \\ \dfrac{\sqrt{3}}{2} & -1/2 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

\[ \sigma_h=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

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

(a)0

(b)-2

(c)1

(d)-1

Answer

Answer: A ✓ checked by 4AB · confidence high

Explanation

S₃² = (σ_h C₃)² = σ_h² C₃² = C₃². Its matrix is a rotation by 240°: trace = 1 + 2cos 240° = 1 − 1 = 0.

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