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GATE Overflow Test Series | Linear Algebra | Test 1 | Question: 27

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The matrix $A=\begin{bmatrix}
3& 2 & 2\\
2 & 3 & 2\\
2& 2 & 3
\end{bmatrix}$ has three distinct eigenvalues and one of its eigenvectors is $\begin{bmatrix}
1\\
1\\
1\\
\end{bmatrix}$. Which one of the following can be another eigenvector of $A$?

  1. $\begin{bmatrix} -1\\
    1\\
    1\end{bmatrix}$
  2. $\begin{bmatrix} 1\\
    -1\\
    1\end{bmatrix}$
  3. $\begin{bmatrix} -1\\
    1\\
    0\end{bmatrix}$
  4. $\begin{bmatrix} -1\\
    0\\
    -1 \end{bmatrix}$
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1 Answer

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The given matrix is a symmetric matrix $(A^{T} = A )$.

For a symmetric matrix, the eigenvectors corresponding to the distinct eigenvalues are always orthogonal (perpendicular).
$$\overrightarrow{x} \cdot \overrightarrow{y} = 0$$
So, the correct answer is $(C).$

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