- #1

whatisreality

- 290

- 1

## Homework Statement

Let A∈M

_{2x2}(ℝ) such that A

^{T}A = I and det(A) = -1. Prove that for ANY such matrix there exists an angle θ such that

A = ##

\left( \begin{array}{cc}

cos(\theta) & sin(\theta)\\

sin(\theta) & -cos(\theta)\\

\end{array} \right) ##

It is not sufficient to show that this matrix satisfies the specified relations.

## Homework Equations

## The Attempt at a Solution

Where do I start with this?! I'm supposed to get from A

^{T}A = I to the rotation matrix! I have looked at several proofs online. But they seem to either use eigenvalues and vectors (which we haven't done, so can't use them!) or don't mention the properties I've been given.

I know a few things that might be useful.

##(A^T)^{-1} = (A^{-1})^T##

And ##det(A^T) = det(A)##

Also, I know that the matrix A is orthogonal.

Don't know how to start!