Linear Algebra, power of matrix [Linear Algebra]Please help. Without using the method of Diagonalization. 

I have tried changing the matrix into 
[sin(pi/6), -tan(pi/6)]
[-tan(pi/6), sin(pi/6)]
and try to figure out what is the linear transformation of A. It seems not to work.
I have also tried to write down the first few power and hope to find the pattern of it, but it doesn't work too.
 A: There are only a few things that can happen in 2D. A symmetric matrix has orthogonal eigenvectors and real eigenvalues, so this is simply a stretch in two orthogonal directions by different amounts.
There are many ways to solve this. For instance, you can write this matrix as a sum:
$$A=\frac12-\frac{1}{\sqrt{3}} \sigma$$
where $\sigma=\begin{bmatrix}0&1\\1&0\end{bmatrix}$.
It follows that $\sigma^2=1$. In fact, the set $\{1,\sigma\}$ is a subgroup of $\mathbb{R}_{2\times 2}$, so all powers and products of such matrices will have the same form.
For instance you have $\det (a+b\sigma)=a^2-b^2$ and
$$(a+b\sigma)^2=a^2+b^2+2ab\sigma$$
$$(a+b\sigma)^{-1}=\frac{1}{a^2-b^2}(a-b\sigma)$$
You can use this to compute anything you want. For nonnegative $n$:
$$(a+b\sigma)^n=\sum {n\choose k}a^{n-k}(b\sigma)^k$$
$$=\sum_{k\in \text{even}} {n\choose k}a^{n-k}b^k+\sigma\sum_{k\in\text{odd}} {n\choose k} a^{n-k}b^k$$
You could also simply guess the results or brute-force the multiplications.
Edit: the group, generated by ${1,\sigma}$ are the split-complex numbers:
http://en.wikipedia.org/wiki/Split-complex_number
EDIT: the above was an idea how to compute all the powers. As I said, this matrix just compresses and stretches along two perpendicular axes. Because of symmetry (components of the vector can be reversed), you know that one eigenvector is along the diagonal $x=y$ (squeezing direction) and the other along the other diagonal (perpendicular to the first one). The transformation is squeezing diagonally. There isn't much more to it. The matrix is nothing special - it doesn't preserve angles or area. I'm not entirely sure if the hint about geometric meaning helps in this case.
