GATE 2023 PH – Question 17

Correct answer: (B) $\begin{pmatrix}\sin2\theta&\cos2\theta\\\cos2\theta&-\sin2\theta\end{pmatrix}$

Explanation

A rank-2 contravariant tensor transforms as $T^\prime=R\,T\,R^T$, where $R$ is the rotation matrix of the coordinate axes. For axes rotated anticlockwise by $\theta$,
$$R=\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}.$$

**Step 1: $RT$** with $T=\begin{pmatrix}0&1\\1&0\end{pmatrix}$:
$$RT=\begin{pmatrix}\sin\theta&\cos\theta\\\cos\theta&-\sin\theta\end{pmatrix}.$$

**Step 2: multiply by $R^T=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$:**
- $(1,1)$: $\sin\theta\cos\theta+\cos\theta\sin\theta=\sin2\theta$
- $(1,2)$: $-\sin^2\theta+\cos^2\theta=\cos2\theta$
- $(2,1)$: $\cos^2\theta-\sin^2\theta=\cos2\theta$
- $(2,2)$: $-\cos\theta\sin\theta-\sin\theta\cos\theta=-\sin2\theta$

$$T^\prime=\begin{pmatrix}\sin2\theta&\cos2\theta\\\cos2\theta&-\sin2\theta\end{pmatrix}\quad(\text{option B}).$$

(Check: the trace is 0 in both frames, as it must be for a tensor transformed by a rotation.)