Is k an Eigenvalue of A with Sum of Row Entries as k?

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SUMMARY

The discussion centers on the question of whether a square matrix A, where the sum of the entries of each row equals a constant k, has k as an eigenvalue. The relevant equation is Ax - λx = 0, leading to the characteristic polynomial (1-k)(k-λ) - k = 0. The inquiry also seeks to identify a corresponding eigenvector that results in the vector (k, k, k, ..., k)T when multiplied by A. The conclusion is that k is indeed an eigenvalue of A, with the corresponding eigenvector being a vector of ones.

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1. Homework Statement

Suppose that A is a square matrix and the sum of the entries of each row is some number k. Is k an eigenvalue of A? if so, what is the corresponding Eigenvector?2. Homework Equations

Ax-λx=0
3. The Attempt at a Solution

(1-k)(K-λ)-k=0I am not sure how to solve this in a proof sense ( since we are not using specific numbers)
 
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Is there a vector can you multiply into A where the product will be (k, k, k, ..., k)T?
 

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