Given the matrices for C₃ and σ_h below,

the trace of the matrix representing for  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  followed by reflection through a plane perpendicular to this axis produces an indistinguishable structure.
• S_n It can be represented by,

• 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,

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,

​

• Now,  is given by,

​

• Thus, trace of the matrix  is given by,
   

Trace of matrix  is,

Conclusion:-

Hence, trace of the matrix representing for  is 0