How do I find the matrix?

Question

Let $\mathbf{A}$ be a matrix and $\mathbf{x}$ be a non-zero vector such that $\mathbf{A} \mathbf{x} = 2 \mathbf{x}.$

Then we have that $\mathbf{A}^7\mathbf{x} = a \mathbf{x}$ some value of $a$. What is $a$ equal to?

I know I need to start by finding "A" but I'm not sure how. And will I need to do $\mathbf{A}^7$ by multiplying "A" seven times or is there a faster way?

Answer 1

$ a = 2^ 7 $ since $2$ is eigenvalue of A so $2^7 $ is an eigenvalue of $A^7$ and be aware that your eigen vector is $x$ in both cases

Answer 2

Observe:

$$ A^2x \;\; =\;\; A(Ax) \;\; =\;\; A(2x) \;\; =\;\; 2(Ax) \;\; =\;\; 4x. $$

Notice a pattern?