https://file.aiquickdraw.com/mathbot/file/1725454081513tou0v62n.jpg

Eigen vector using cross multiplication method detailed steps

To find eigenvectors using the cross-multiplication method, we begin by calculating the characteristic polynomial of the matrix \( A \) and then use the given eigenvalues to find corresponding eigenvectors.

### Given Matrix and Information
The given matrix \( A \) is:
\[
A = \begin{bmatrix} 3 & 10 & 5 \\ -2 & -3 & -4 \\ -3 & -5 & 1 \end{bmatrix}
\]

The eigenvalues given are \( \lambda = -3, 2, 2 \).

We will use one of these eigenvalues, let's start with \( \lambda = 2 \), to find the corresponding eigenvectors.

### Steps to Find the Eigenvector using Cross-Multiplication Method

1. **Substitute the Eigenvalue in the Matrix:**
   The eigenvalue equation is \( (A - \lambda I) \mathbf{x} = 0 \), where \( I \) is the identity matrix.

   For \( \lambda = 2 \):
   \[
   A - 2I = \begin{bmatrix} 3 & 10 & 5 \\ -2 & -3 & -4 \\ -3 & -5 & 1 \end{bmatrix} - 2 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 10 & 5 \\ -2 & -5 & -4 \\ -3 & -5 & -1 \end{bmatrix}
   \]

2. **Set Up the System of Equations:**
   The system of equations \( (A - 2I) \mathbf{x} = 0 \) can be written as:
   \[
   \begin{aligned}
   1x_1 + 10x_2 + 5x_3 &= 0, \\
   -2x_1 - 5x_2 - 4x_3 &= 0, \\
   -3x_1 - 5x_2 - x_3 &= 0.
   \end{aligned}
   \]

3. **Use Cross-Multiplication Method to Find Relationships:**
   To find the eigenvector, you can use cross multiplication on the coefficients of the equations. The cross product helps find proportional relationships between \( x_1, x_2, x_3 \). 
   
   Let's set two equations:
   \[
   \text{Equation 1: } \begin{vmatrix} -5 & -4 \\ -5 & -1 \end{vmatrix} = (-5)(-1) - (-4)(-5) = 5 + 20 = 25,
   \]
   \[
   \text{Equation 2: } \begin{vmatrix} -2 & -4 \\ -3 & -1 \end{vmatrix} = (-2)(-1) - (-4)(-3) = 2 + 12 = 14,
   \]
   \[
   \text{Equation 3: } \begin{vmatrix} -2 & -5 \\ -3 & -5 \end{vmatrix} = (-2)(-5) - (-5)(-3) = 10 + 15 = 25.
   \]

4. **Find the Eigenvector:**
   Set up the ratios:
   \[
   \frac{x_1}{25} = \frac{x_2}{14} = \frac{x_3}{25}.
   \]

   Simplify to get the vector form:
   \[
   \mathbf{x} = k \begin{bmatrix} 25 \\ 14 \\ 25 \end{bmatrix},
   \]
   where \( k \) is any scalar.

This method shows the detailed steps of using cross-multiplication to derive eigenvectors. You can verify this eigenvector with other equations to ensure it holds true.

Would you like more details on these calculations or have any specific questions?

### Further Questions:
1. How can we find the eigenvector for \( \lambda = -3 \) using this method?
2. What is the significance of the eigenvectors in the context of the given matrix?
3. How can the cross-multiplication method be applied to other eigenvalues?
4. What are alternative methods for finding eigenvectors?
5. How do the properties of the matrix affect the calculation of eigenvectors?

**Tip:** Always double-check your calculations when using the cross-multiplication method to ensure consistency, especially with signs and arithmetic.

Explore how to find eigenvectors using the cross-multiplication method with a step-by-step guide. This example includes eigenvalues λ = -3 and λ = 2, applicable for linear algebra studies. Perfect for undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation