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To find the eigenvalues of matrix \( A \), we need to solve the characteristic equation given by:

\[
\det(A - \lambda I) = 0
\]

where \( A \) is the matrix:

\[
A = \begin{bmatrix}
0 & 0 & 2 \\
0 & 2 & 0 \\
0 & 0 & -1
\end{bmatrix},
\]

and \( I \) is the 3x3 identity matrix:

\[
I = \begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}.
\]

Now, the matrix \( A - \lambda I \) is:

\[
A - \lambda I = \begin{bmatrix}
0 - \lambda & 0 & 2 \\
0 & 2 - \lambda & 0 \\
0 & 0 & -1 - \lambda
\end{bmatrix}
=
\begin{bmatrix}
-\lambda & 0 & 2 \\
0 & 2 - \lambda & 0 \\
0 & 0 & -1 - \lambda
\end{bmatrix}.
\]

The characteristic equation is found by calculating the determinant of \( A - \lambda I \):

\[
\det(A - \lambda I) = \left(-\lambda\right) \left( (2 - \lambda)(-1 - \lambda) \right) - (0) + (0).
\]

Simplify the determinant:

\[
\det(A - \lambda I) = -\lambda \left( (2 - \lambda)(-1 - \lambda) \right).
\]

Expand the determinant:

\[
(2 - \lambda)(-1 - \lambda) = -2 - 2\lambda + \lambda + \lambda^2 = \lambda^2 - \lambda - 2.
\]

So,

\[
\det(A - \lambda I) = -\lambda(\lambda^2 - \lambda - 2).
\]

To find the eigenvalues, set the determinant to zero:

\[
-\lambda(\lambda^2 - \lambda - 2) = 0.
\]

This equation implies:

1. \( \lambda = 0 \)
2. Solve \( \lambda^2 - \lambda - 2 = 0 \)

The quadratic equation can be factored as:

\[
(\lambda - 2)(\lambda + 1) = 0.
\]

Therefore, the solutions are:

\[
\lambda = 2, \, \lambda = -1.
\]

### Eigenvalues of Matrix \( A \)

The eigenvalues of matrix \( A \) are:

\[
\lambda_1 = 0, \, \lambda_2 = 2, \, \lambda_3 = -1.
\]

Would you like more details on the steps or explanations for finding eigenvectors? Here are some related questions that you might find helpful:

1. How do you find the eigenvectors corresponding to each eigenvalue?
2. What is the significance of eigenvalues and eigenvectors in linear algebra?
3. Can eigenvalues be complex for real matrices?
4. What does the determinant of a matrix tell us about its eigenvalues?
5. How does the trace of a matrix relate to its eigenvalues?

**Tip:** Remember that the sum of the eigenvalues of a matrix is equal to its trace, and the product of the eigenvalues is equal to its determinant.

Find the eigenvalues of matrix A, where A is a 3x3 matrix given by: 
A = [ [ 0, 0, 2 ], [ 0, 2, 0 ], [ 0, 0, -1 ] ]

Learn how to find the eigenvalues of a 3x3 matrix. This example involves solving the characteristic equation, using the determinant of (A - λI), and applying the quadratic formula. Suitable for undergraduate students learning linear algebra.

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