in Linear Algebra recategorized by
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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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