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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.

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  1. Loser66
    • one year ago
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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?

  2. ganeshie8
    • one year ago
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    It is not hard to see that the matrix will be of form \[\begin{bmatrix} a&b\\b&a\end{bmatrix}\]

  3. Loser66
    • one year ago
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    Yes, sir

  4. ganeshie8
    • one year ago
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    Next i would multiply each of the given eigen vectors and see if any makes sense

  5. Loser66
    • one year ago
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    Yes, just III, right?

  6. ganeshie8
    • one year ago
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    Yep!

  7. Loser66
    • one year ago
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    Hence, only one way!!! right? do steps, right?

  8. ganeshie8
    • one year ago
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    \[\begin{bmatrix} a&b\\b&a\end{bmatrix} \begin{bmatrix}1\\0\end{bmatrix}~~ \stackrel{?}{=}~~ \lambda \begin{bmatrix} 1\\0\end{bmatrix}\]

  9. Loser66
    • one year ago
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    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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