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## Homework Statement

Let A be

*n x n, λ ∈ ℂ,*let

*v*be

*n x 1*, and suppose that A ⋅

*v = λ ⋅*v. Show that A

*^j*⋅

*v =
^j ⋅ v*for each positive integer j.

## Homework Equations

## The Attempt at a Solution

I haven't been able to get very far but,

*- A
^j ⋅ v*

*^j*⋅

*v = 0*

_{n x 1}*v(*A

*
^j - **^j) = 0*

_{n x 1}Not sure how to prove that for every positive j that this is true. Any thought would be appreciated. Thanks.