Seeing what matrices do geometrically I am somewhat able to identify what simple matrices do geometrically once i have a look at the matrix itself. However I am not too sure how to tackle this when cos(α) , sin(α), -sin(α) are part of the matrix.
 A: Since every matrix can be viewed as a linear transformation, there a variety of ways to see what it may do to vectors geometrically. 
The easiest way (as has been suggested) is to find how it operates on the standard basis vectors and investigate what it does to each of them (write the coordinate matrices, since the image of the transformation is completely determined by the bases of its domain and codomain).
Another way to analyze matrices is to find its eigenvalues and eigenvectors, and you characterize a large span of vectors from this (which vectors are preserved and only distance is changed). 
Some special cases (as you have mentioned) are matrices that are involved in a symmetry action. Orthogonal matrices act as isometries, either rotations and reflections. You can find this by investigating its determinant. In general, $detA = 1$ corresponds to a rotation while $detA = -1$ corresponds to a reflection. If its determinant is $1$, you have a rotation matrix that rotates about the plane normal to the vector $v$ corresponding to $\lambda = 1$.  
There are probably many more cases that might yield not very interesting results. 
A: If you can multiply matricies by vectors, then you can see what a transformation does. For example, the matrix $T=\left(\matrix{0 & 1 \\ 1 & 0}\right)$, take the vector $\left(\matrix{1 \\ 0}\right)$ and multiply. You will get another vector. This is the result of the transformation $T$ on the first basis vector. Similarly, Multiply $T$ by $\left(\matrix{0 \\ 1}\right)$. This is the effect of the transformation $T$.
In other words, $T$ switches the $x$ and $y$ coordinates of any vector $x$. 
To see what your matrix does, just multiply it by the basis vectors. It may help to plug in an angle and draw the resulting vectors in $\mathbb{R}^2$. 
