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We are given a matrix \( A = \begin{bmatrix} 0 & 3 \\ -3 & 0 \end{bmatrix} \) and need to determine which vector from the provided options is an eigenvector of this matrix.

### Steps to determine the eigenvector:
1. To be an eigenvector, a vector \( v \) must satisfy the equation \( A \cdot v = \lambda \cdot v \), where \( \lambda \) is the eigenvalue, and \( v \) is the eigenvector.
2. First, find the eigenvalues \( \lambda \) by solving the characteristic equation:
   \[
   \text{det}(A - \lambda I) = 0
   \]
   where \( I \) is the identity matrix.

   We compute the determinant:
   \[
   \text{det} \begin{bmatrix} -\lambda & 3 \\ -3 & -\lambda \end{bmatrix} = \lambda^2 + 9 = 0
   \]
   Solving \( \lambda^2 + 9 = 0 \) gives \( \lambda = \pm 3i \).

3. Now, find the eigenvectors corresponding to \( \lambda = 3i \) and \( \lambda = -3i \).

### Eigenvectors for \( \lambda = 3i \):
We solve \( (A - 3iI)v = 0 \). This leads to a system of linear equations, and solving it yields the eigenvector corresponding to \( \lambda = 3i \), which involves complex numbers like \( 1 + i \) or \( 1 - i \).

### Conclusion:
From the list of options, the vector \( \begin{bmatrix} 1 + i \\ i \end{bmatrix} \) matches an eigenvector associated with the eigenvalue \( 3i \). Therefore, the correct answer is the second option: \( \begin{bmatrix} 1 + i \\ i \end{bmatrix} \).

Would you like more details on any step? Here are related questions to deepen your understanding:

1. How do you compute the determinant of a 2x2 matrix?
2. What are the properties of complex eigenvalues and their eigenvectors?
3. Can a matrix have both real and complex eigenvalues?
4. How do you solve for eigenvectors once eigenvalues are found?
5. Why does the characteristic polynomial have complex roots in this case?

**Tip:** When dealing with complex eigenvalues, remember that their corresponding eigenvectors will also involve complex components.

Which of the following is an eigenvector of the matrix [ 0  3 ; -3  0 ]?

Learn how to determine which vector is an eigenvector of the matrix [ 0  3 ; -3  0 ]. The problem involves solving the characteristic equation and finding eigenvalues, followed by calculating the corresponding eigenvectors. Suitable for undergraduate linear algebra students.

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