Geometric interpretation of a matrix I'm supposed to be able to determine without calculations the determinant, inverse matrix, and n-th power matrix of the rotation matrix :
$\begin{pmatrix} 
cos\theta & sin\theta  \\ 
-sin\theta & cos\theta 
\end{pmatrix} $
Can someone explain to me how I can do that ?
 A: HINTS
The determinant tells you by how much a linear transformation transforms areas (for $2\times 2$)/ volumes (for $3\times 3$)/ etc.  So by what factor does this transformation change the area of say a square in the plane?  That'll be your determinant.
The inverse matrix is the matrix which does the inverse transformation.  What exactly does this matrix do?  How could that action be undone?
If you applied this matrix to a vector more than once, what should happen?
A: The matrices you wrote is a rotation in the plane by an angle of $\theta$. Therefore its inverse can be obtained by replacing $\theta$ by $-\theta$ as a rotation can be undone by a rotation with the opposite angle. Similarly its $n$th power will be a rotation by the angle $n\theta$. Finally as the determinant measures volumes and a rotation is an isometry (more precisely if two vectors $u$, $v$ determine a parallelepiped of volume $V$, and $T$ is a linear transformation, then the volume of the parallelepiped generated by $Tu$, $Tv$ is $det(T)\cdot V$) the determinant is one.
