Loser66
  • Loser66
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
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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Loser66
  • 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?
ganeshie8
  • ganeshie8
It is not hard to see that the matrix will be of form \[\begin{bmatrix} a&b\\b&a\end{bmatrix}\]
Loser66
  • Loser66
Yes, sir

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ganeshie8
  • ganeshie8
Next i would multiply each of the given eigen vectors and see if any makes sense
Loser66
  • Loser66
Yes, just III, right?
ganeshie8
  • ganeshie8
Yep!
Loser66
  • Loser66
Hence, only one way!!! right? do steps, right?
ganeshie8
  • ganeshie8
\[\begin{bmatrix} a&b\\b&a\end{bmatrix} \begin{bmatrix}1\\0\end{bmatrix}~~ \stackrel{?}{=}~~ \lambda \begin{bmatrix} 1\\0\end{bmatrix}\]
Loser66
  • Loser66
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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