find eigenvalues and eigenvectors of [7 0 "-3" "-9" "-2" 3 18 0 "-8]"

Let's find the eigenvalues and eigenvectors of the matrix:

\[
A = \begin{pmatrix}
7 & 0 & -3 \\
-9 & -2 & 3 \\
18 & 0 & -8
\end{pmatrix}
\]

### Step 1: Find the Eigenvalues
The eigenvalues of a matrix are found by solving the characteristic equation:

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

Where \( \lambda \) is the eigenvalue and \( I \) is the identity matrix.

For matrix \( A \), we have:

\[
A - \lambda I = \begin{pmatrix}
7 - \lambda & 0 & -3 \\
-9 & -2 - \lambda & 3 \\
18 & 0 & -8 - \lambda
\end{pmatrix}
\]

The determinant is:

\[
\text{det}(A - \lambda I) = \text{det}\left(\begin{pmatrix}
7 - \lambda & 0 & -3 \\
-9 & -2 - \lambda & 3 \\
18 & 0 & -8 - \lambda
\end{pmatrix}\right)
\]

We will calculate this determinant to find the characteristic polynomial and then solve for \( \lambda \) (the eigenvalues).

### Step 2: Solve for Eigenvectors
Once the eigenvalues are found, we solve \( (A - \lambda I) \mathbf{v} = 0 \) to find the corresponding eigenvectors.

Let's calculate these.### Results:

1. **Eigenvalues**: The eigenvalues of the matrix \( A \) are:
   \[
   \lambda_1 = -2, \quad \lambda_2 = 1, \quad \lambda_3 = -2
   \]

2. **Eigenvectors**: The corresponding eigenvectors are:

   For \( \lambda_1 = -2 \):
   \[
   \mathbf{v}_1 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}
   \]

   For \( \lambda_2 = 1 \):
   \[
   \mathbf{v}_2 = \begin{pmatrix} 0.408 \\ -0.408 \\ 0.816 \end{pmatrix}
   \]

   For \( \lambda_3 = -2 \):
   \[
   \mathbf{v}_3 = \begin{pmatrix} 0.300 \\ -0.318 \\ 0.899 \end{pmatrix}
   \]

These results give you the eigenvalues and the corresponding eigenvectors for the matrix \( A \).

Would you like more details on any part of the solution, or do you have any questions?

### Related Questions:
1. How are eigenvalues and eigenvectors used in solving systems of differential equations?
2. What is the geometric interpretation of eigenvectors and eigenvalues?
3. How can the diagonalization of a matrix be related to its eigenvalues and eigenvectors?
4. What happens if a matrix has complex eigenvalues?
5. How can eigenvectors be normalized?

### Tip:
When computing eigenvalues and eigenvectors, always check if the matrix is diagonalizable to simplify your calculations.

Learn how to find eigenvalues and eigenvectors of a 3x3 matrix with detailed step-by-step solutions. This problem covers key concepts including eigenvalues, eigenvectors, matrix determinants, and characteristic equations. Suitable for advanced high school students and beyond.

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