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GATE DA 2026 Q46: Rotation angle from the trace of a 3×3 rotation matrix

Question

Let γ1,γ2,γ3\gamma_1, \gamma_2, \gamma_3 be the eigenvalues of the matrix [1000cos⁡tsin⁡t0−sin⁡tcos⁡t]\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 & -\sin t & \cos t \end{bmatrix}, where t∈[−π,π]t \in [-\pi, \pi] is in radians.

Which one of the following options lists all the possible values of tt satisfying γ1+γ2+γ3=1+2\gamma_1 + \gamma_2 + \gamma_3 = 1 + \sqrt{2}?

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Answer: C (official GATE 2026 final answer key)

Concept used: The sum of the eigenvalues equals the trace.

First step: Trace =1+2cos⁡t=1+2= 1 + 2\cos t = 1 + \sqrt{2}, so cos⁡t=12\cos t = \tfrac{1}{\sqrt{2}}.

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Source: official GATE 2026 DA question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key

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