Loser66 one year ago 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.

1. Loser66

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?

2. ganeshie8

It is not hard to see that the matrix will be of form $\begin{bmatrix} a&b\\b&a\end{bmatrix}$

3. Loser66

Yes, sir

4. ganeshie8

Next i would multiply each of the given eigen vectors and see if any makes sense

5. Loser66

Yes, just III, right?

6. ganeshie8

Yep!

7. Loser66

Hence, only one way!!! right? do steps, right?

8. ganeshie8

$\begin{bmatrix} a&b\\b&a\end{bmatrix} \begin{bmatrix}1\\0\end{bmatrix}~~ \stackrel{?}{=}~~ \lambda \begin{bmatrix} 1\\0\end{bmatrix}$

9. Loser66

if a = lambda and b = 0 but if it is, then the $$A - I\lambda$$ is a zero matrix--> no more lambda