GATE 2020 PH – Question 42

Mathematical Physics · Matrices: similarity transformations, diagonalization, eigenvalues and eigen vectors · 2 marks · Multiple choice

Which matrix does NOT represent a proper rotation in a plane?

$$\begin{pmatrix}-\sin\theta&\cos\theta\\-\cos\theta&-\sin\theta\end{pmatrix}$$
$$\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}$$
$$\begin{pmatrix}\sin\theta&\cos\theta\\-\cos\theta&\sin\theta\end{pmatrix}$$
$$\begin{pmatrix}-\sin\theta&\cos\theta\\-\cos\theta&\sin\theta\end{pmatrix}$$

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Correct answer: (D) $$\begin{pmatrix}-\sin\theta&\cos\theta\\-\cos\theta&\sin\theta\end{pmatrix}$$

Explanation

A proper two-dimensional rotation is orthogonal with determinant +1. The first three matrices have orthonormal columns and determinant $\sin^2\theta+\cos^2\theta=1$. D instead has determinant $\cos^2\theta-\sin^2\theta$, which is not identically 1, and its columns are not generally orthogonal. Hence D does not give a proper rotation for arbitrary θ.

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