Let A be a 2x2 matrix for which there is a constant k such that the sum of the entries in each row and each column is k. Which of the following must be an eigenvector of A? I) (1,0) II) (0,1) III)(1,1) A) I only B) II only C) III only D) I, II E)I,II,III Please, help.

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Let A be a 2x2 matrix for which there is a constant k such that the sum of the entries in each row and each column is k. Which of the following must be an eigenvector of A? I) (1,0) II) (0,1) III)(1,1) A) I only B) II only C) III only D) I, II E)I,II,III Please, help.

Mathematics
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As usual, I would like to know there is any shortcut to find the answer or I have to go step by step to find eigenvectors of A?
It is not hard to see that the matrix will be of form \[\begin{bmatrix} a&b\\b&a\end{bmatrix}\]
Yes, sir

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Next i would multiply each of the given eigen vectors and see if any makes sense
Yes, just III, right?
Yep!
Hence, only one way!!! right? do steps, right?
\[\begin{bmatrix} a&b\\b&a\end{bmatrix} \begin{bmatrix}1\\0\end{bmatrix}~~ \stackrel{?}{=}~~ \lambda \begin{bmatrix} 1\\0\end{bmatrix}\]
if a = lambda and b = 0 but if it is, then the \(A - I\lambda\) is a zero matrix--> no more lambda

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