https://file.aiquickdraw.com/mathbot/file/1727355306617z0cvmj0g.jpg

### Step-by-Step Solution for Eigenvalues and Eigenvectors

1. **Given Matrix \( A \):**

\[
A = \begin{pmatrix}
-1 & -2 & -3 \\
4 & 5 & -6 \\
7 & -8 & 9
\end{pmatrix}
\]

2. **Eigenvalues:**
   The characteristic equation is derived from \( \det(A - \lambda I) = 0 \), where \( \lambda \) represents the eigenvalues. The eigenvalues of the matrix \( A \) are solutions to this equation.

   The eigenvalues for this matrix are complex, expressed as:

\[
\lambda_1 = 13/3 + (-1/2 - \frac{\sqrt{3} i}{2}) \cdot \alpha + \frac{133}{9 \cdot (-1/2 - \frac{\sqrt{3} i}{2}) \cdot \alpha}
\]
\[
\lambda_2 = 13/3 + \frac{133}{9 \cdot (-1/2 + \frac{\sqrt{3} i}{2}) \cdot \alpha} + (-1/2 + \frac{\sqrt{3} i}{2}) \cdot \alpha
\]
\[
\lambda_3 = \frac{133}{9 \cdot \alpha} + 13/3 + \alpha
\]

Where \( \alpha = (2 \cdot \sqrt{13043} + \frac{6355}{27})^{1/3} \).

3. **Eigenvectors:**
   Each eigenvalue has a corresponding eigenvector. For each \( \lambda \), we solve the system \( (A - \lambda I) \mathbf{v} = 0 \) to find the eigenvector \( \mathbf{v} \).

   The eigenvectors corresponding to the eigenvalues are expressed as matrices, each with complex entries:

   - For \( \lambda_1 \):
     \[
     \mathbf{v}_1 = \begin{pmatrix}
     \frac{2}{5} + \dots \\
     \frac{14}{15} + \dots \\
     1
     \end{pmatrix}
     \]

   - For \( \lambda_2 \):
     \[
     \mathbf{v}_2 = \begin{pmatrix}
     \frac{2}{5} + \dots \\
     \frac{14}{15} + \dots \\
     1
     \end{pmatrix}
     \]

   - For \( \lambda_3 \):
     \[
     \mathbf{v}_3 = \begin{pmatrix}
     \frac{2}{5} + \dots \\
     \frac{14}{15} + \dots \\
     1
     \end{pmatrix}
     \]

### Summary:
- The eigenvalues are complex numbers with real and imaginary parts.
- The eigenvectors are complex as well, with expressions depending on the cube root term \( \alpha \).

Would you like further details on any of the steps?

### 5 Related Questions:
1. How do we simplify cube roots in eigenvalue calculations?
2. What is the significance of complex eigenvalues in this matrix?
3. How are eigenvectors geometrically interpreted?
4. How do eigenvalues relate to the determinant and trace of a matrix?
5. What applications can matrices with complex eigenvalues have?

### Tip:
When working with complex eigenvalues and eigenvectors, it is important to carefully track real and imaginary parts to avoid errors in interpretation.

Find the eigenvalues and eigenvectors of the matrix A = [[-1, -2, -3], [4, 5, -6], [7, -8, 9]].

Learn how to find the eigenvalues and eigenvectors of the matrix A = [[-1, -2, -3], [4, 5, -6], [7, -8, 9]] with a step-by-step solution. This problem involves solving the characteristic equation and finding the corresponding eigenvectors. Suitable for undergraduate-level linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation