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GATE 2017 EE – Question 11

Engineering Mathematics · Linear Algebra: Eigen values, Eigen vectors · 1 mark · Multiple choice

The matrix $A = \begin{bmatrix} \frac{3}{2} & 0 & \frac{1}{2} \\ 0 & -1 & 0 \\ \frac{1}{2} & 0 & \frac{3}{2} \end{bmatrix}$ has three distinct eigenvalues and one of its eigenvectors is $\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$. Which one of the following can be another eigenvector of $A$?

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

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Correct answer: (C) $\begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}$

Explanation

The matrix is symmetric, so eigenvectors for different eigenvalues are perpendicular. A vector perpendicular to $(1, 0, 1)$ is $(1, 0, -1)$, and $A(1, 0, -1)^T = \left(\frac{3}{2} - \frac{1}{2}, 0, \frac{1}{2} - \frac{3}{2}\right) = (1, 0, -1)$, so it is an eigenvector with eigenvalue 1. The other options do not satisfy $A\vec{v} = \lambda\vec{v}$.