1. ## Rotation matrix.

Explain what it means for a matrix $M$ to be orthogonal.
With respect to the standard basis in cartesian plane, the anticlockwise rotation through angle $\theta$ around the origin is represented by:

${M}_{\theta }$ =

( $\cos \theta$ $-\sin \theta$)
( $\sin \theta$ $\cos \theta$)

Supposed to be 2x2 matrix sorry I don't know the tex for it?

Verify that ${M}_{\theta }$ is orthogonal and verify the identity ${M}_{\theta + \varphi} = {M}_{\theta}{M}_{\varphi}$

2. Originally Posted by TeaWithoutMussolini
Explain what it means for a matrix $M$ to be orthogonal.
With respect to the standard basis in cartesian plane, the anticlockwise rotation through angle $\theta$ around the origin is represented by:

${M}_{\theta }$ =

( $\cos \theta$ $-\sin \theta$)
( $\sin \theta$ $\cos \theta$)

Supposed to be 2x2 matrix sorry I don't know the tex for it?

Verify that ${M}_{\theta }$ is orthogonal and verify the identity ${M}_{\theta + \varphi} = {M}_{\theta}{M}_{\varphi}$
I will find all the orthogonal matrices at $M_2(\mathbb{R})$.

Let $A=\begin{pmatrix}a & b\\ c & d\end{pmatrix}$ orthogonal matrix .

Hence, $AA^*=AA^T=I$.

In other words:

$A^T=A^{-1}=\frac{1}{|A|}\begin{pmatrix}d & -b\\ -c & a\end{pmatrix}.$

Now, from $AA^T=I$ we get:

$1=|I|=|AA^T|=|A||A^T|=A^2$

Hence $|A|=\pm1$ .

Suppose that $|A|=1$, then:

$\begin{pmatrix}a & c\\ b & d\end{pmatrix}=\begin{pmatrix}d & -b\\ -c & a\end{pmatrix}$

So: $b=-c, a=d$.

$A^{+}=\begin{pmatrix}a & b\\ -b & a\end{pmatrix}
$

In the case in which $|A|=-1$ we will get:

$\begin{pmatrix}a & c\\ b & d\end{pmatrix}=\begin{pmatrix}-d & b\\ c & -a\end{pmatrix}
$

So: $b=s, a=-d$

$A^{-}=\begin{pmatrix}a & b\\ b & -a\end{pmatrix}$

In both cases we can deduce from orthogonality that $a^2+b^2=1$.

$|a|\leq 1$ so exists angel $0\leq \theta \leq 2\pi$ for which $a=cos\theta$ and then $b=sin\theta$.

Finally we can write he follwing:

$A^+=\begin{pmatrix}cos\theta & sin \theta \\-sin\theta & cos\theta\end{pmatrix}$

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

3. For the first bit here is a link. Orthogonal matrix - Wikipedia, the free encyclopedia

The LaTex code is

\begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \\ \end{bmatrix}

For the last part verify the definitions that you have or the one's in the link and I would start with the left hand side of the equality. Use matrix multiplication and then use the sum identities for sine and cosine.

Good luck