Let A be a matrix and x be a non-zero vector such that Ax=2x. Then we have that A^7x=ax some value of a. What is a equal to?

Bierlehre59

Bierlehre59

Answered question

2022-08-06

Let A be a matrix and x be a non-zero vector such that A x = 2 x .
Then we have that A 7 x = a 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 v by multiplying "A" seven times or is there a faster way?

Answer & Explanation

Isabella Rocha

Isabella Rocha

Beginner2022-08-07Added 10 answers

Observe:
A 2 x = A ( A x ) = A ( 2 x ) = 2 ( A x ) = 4 x .
there is a pattern!
cofak48

cofak48

Beginner2022-08-08Added 2 answers

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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