# How to find a real orthogonal matrix of determinant $1$?

A real orthogonal matrix $A$ is proper if $\det A=1$.

Find $2\times 2$ proper matrix $A$

I tried to use the fact that product of $A$ and its transpose is equal to identity.

But, there were bunch of equations which seem not related to each other and can not find such $A$.

• Example: $2\times 2$ identity. – vadim123 Nov 6 '14 at 22:48

For any $\theta$, $$\begin{pmatrix} \cos\theta & \sin\theta\\ -\sin\theta & \cos\theta \end{pmatrix}$$ is such a matrix. Actually, they all have this form.
• try for instance to substract $\pi / 2$ to $\theta$. – mookid Nov 6 '14 at 23:09