Properties of the rotation matrix $$R(\theta) = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$$
Consider the above rotation matrix.
$a)$ Show that $R(\theta)$ is non-singular with: $R^{-1}(\theta) = R(-\theta)$
$b)$ Show that $R(\theta)^T = R(-\theta)$
$c)$ For what angles $\theta \in \mathbb{R}$ is $R(\theta)$ symmetric?
I only somewhat understand part $b$. I know that a skew-matrix is one in which its transpose is also its negative. Apart from that I'm not sure how to implement that into the question and I have no idea how to do part $a$ and $c$. I was wondering if someone could help me get started.
 A: Hint:
(a) $\quad$ Calculate $R(-\theta)R(\theta)$
(c) $\quad$ By definition: symmetric when $R(\theta)^T=R(\theta)$
$\quad \quad \,$ By $(b)$ then $R(\theta)=R(-\theta) \implies \sin \theta=-\sin \theta$
A: a
A quick singularity check:
$$
\det \mathbf{R} \left( \theta \right) 
= \cos^{2} \left( \theta \right) + \sin^{2} \left( \theta \right) = 1
$$
Because the determinant is not $0$, the rotation matrix is not singular.
Is 
$$
 \mathbf{R} \left( \theta \right)^{-1} = \mathbf{R} \left( -\theta \right)?
$$
Check via multiplication
$$
\begin{align}
  \mathbf{R} \left( \theta \right) \mathbf{R} \left( -\theta \right) &=
%
\left[
\begin{array}{rr}
 \cos ( \theta ) & -\sin (\theta ) \\
 \sin (\theta ) & \cos (\theta ) \\
\end{array}
\right]
%
\left[
\begin{array}{rr}
 \cos (\theta ) & \sin (\theta ) \\
 -\sin (\theta ) & \cos (\theta ) \\
\end{array}
\right] \\[5pt]
%
&=
%
\left[
\begin{array}{cc}
 \cos ^2(\theta )+\sin ^2(\theta ) & 0 \\
 0 & \cos ^2(\theta )+\sin ^2(\theta ) \\
\end{array}
\right]
 \\[5pt]
%
&=
%
\left[
\begin{array}{cc}
 1 & 0 \\
 0 & 1 \\
\end{array}
\right] \\[2pt]
%
&= \mathbf{I}_{2}
%
\end{align}
$$
b
Show that the transpose matrix is $\mathbf{R}\left( -\theta \right)$.
$$
\begin{align}
  \mathbf{R}\left( -\theta \right) &=
\left[
\begin{array}{rr}
 \cos (\theta ) & \sin (\theta ) \\
 -\sin (\theta ) & \cos (\theta ) \\
\end{array}
\right] \\[3pt]
%
  \mathbf{R}^{T}\left( \theta \right) &=
\left[
\begin{array}{rr}
 \cos (\theta ) & \sin (\theta ) \\
 -\sin (\theta ) & \cos (\theta ) \\
\end{array}
\right] \\
\end{align}
%
$$
c
When does $\mathbf{R} \left( \theta \right)$ have a symmetric form like
$$
\left[
\begin{array}{rr}
 a & b \\
 b & b \\
\end{array}
\right] ?
$$
The diagonal entries are the same, $\cos \left( \theta \right)$. They pose no constraint. For the off-diagonal entries, the question is when does
$$
  \sin \left( \theta \right) = - \sin \left( \theta \right)?
$$
The answer is when $\sin \left( \theta \right) = 0?$ which occurs for
$$
  \theta = 2k\pi, \quad k\in\mathbb{Z}.
$$
