Difficulty understanding discrete dynamical systems problem

Question

I'm learning discrete dynamical systems right now in linear algebra and I am running up against an error in one particular problem that I can't seem to solve.

Problem:

Let x(t) = $ \begin{bmatrix} x_1(t)\\ x_2(t)\\ \end{bmatrix} $ be the solution to a discrete dynamical system:

$x_1(t+1)=10x_1(t)+4x_2(t)$

$x_2(t+1)=−8x_1(t)−2x_2(t)$

If x(0) =$ \begin{bmatrix} 5 \\ -3 \\ \end{bmatrix}$.

Put the eigenvalues in ascending order when you enter $x_1(t), x_2(t)$ below:

So far I have found the eigenvalues, but an struggling to find the constants for the equations(as represented by "(blank)". Any help or pointers in the right direction would be greatly appreciated.

Answer so Far:

$x_1=(blank)(2)^t+(blank)(6)^t$

$x_2=(blank)(2)^t+(blank)(6)^t$

Answer 1

With

$$ M = \left( \begin{array}{cc} 10 & 4 \\ -8 & -2 \\ \end{array} \right),\ \ \ Q = \left( \begin{array}{cc} -1 & 1 \\ -1 & 2 \\ \end{array} \right),\ \ \ \Lambda = \left( \begin{array}{cc} 6 & 0 \\ 0 & 2 \\ \end{array} \right) $$

we have

$$ x_{k+1} = Q\cdot\Lambda\cdot Q^{-1}x_k,\ \ \ M = Q\cdot\Lambda\cdot Q^{-1} $$

or

$$ Q^{-1}x_{k+1} = \Lambda\cdot Q^{-1}x_k $$

or

$$ X_{k+1} = \Lambda\cdot X_k $$

with solution

$$ X_k = \Lambda^{k-1}X_0 $$

and finally

$$ x_k = Q\cdot \Lambda^{k-1}Q^{-1}x_0 $$

Answer 2

If you have difficulties with the eigenvector approach, you can also just simply compute $x(1)=\begin{bmatrix}38\\-34\end{bmatrix}$ and solve the two systems $$ 5=a_1+a_2\\ 38=2a_1+6a_2 $$ for the coefficients of the $x_1$ component and $$ -3=b_1+b_2\\ -34=2b_1+6b_2 $$ for the $x_2$ component.