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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?

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2 Answers 2

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Observe:

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

Notice a pattern?

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  • $\begingroup$ Thank you, but how did you go from A(Ax) to A(2x)? $\endgroup$
    – hailey
    Apr 9, 2020 at 17:07
  • $\begingroup$ @hailey That was given in your prompt. $x$ is a vector such that $Ax = 2x$, is it not? $\endgroup$
    – Mnifldz
    Apr 9, 2020 at 17:08
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$ 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

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