Let $A(\theta)$ be a given function , where $\theta \in (0, 2\pi)$. Mark the correct statement below

Question

Let $ A(θ) = \left[ {\begin{array}{cc} \cosθ & \sinθ \\ -\sinθ & \cosθ \\ \end{array} } \right] $ where $θ ∈ (0, 2π)$. Mark the correct statement below

A. $A(θ)$ has eigenvectors in $\mathbb R^2$ for all $θ ∈ (0, 2π)$

B. $A(θ)$ does not have an eigenvector in $\mathbb R^ 2$ , for any $θ ∈ (0, 2π)$

C. $A(θ)$ has eigenvectors in $\mathbb R^ 2$ , for exactly one value of $θ ∈ (0, 2π)$

D. $A(θ)$ has eigenvectors in $\mathbb R^ 2$ , for exactly 2 values of $θ ∈ (0, 2π)$

A. $\theta=\frac{\pi}{2}$, statement is false.

B. $\theta=\pi$, $A=-\mathbb I$, all the vectors in $\mathbb R$ are eigen vectors. So statement is false.

I am not able to judge (C) and (D). how to judge it? Please help me.

Answer 1

The matrix is clockwise rotation by $\theta$ (or equivalently, counterclockwise rotation by $-\theta$).

If you rotate a nonzero vector clockwise by an angle $\theta$, the only way the result can be parallel to the original is if $\theta$ is a multiple of $\pi$.

But it's given that $\theta \in (0,2\pi)$, so . . .

Answer 2

The answer is C. You can actually compute the Eigenvalues and see. But you can also think about this: by what angle can I rotate a vector so that I end up with a real multiple of the vector. It’s clear that 0, π and 2 π are the only choices, but only one of them lies in the range (0,2 π )