Find eigen value and eigen vector of the matrix 2 3 4 3 1 2 -1 2 2

We are given the matrix \( A \):

\[
A = \begin{pmatrix}
2 & 3 & 4 \\
3 & 1 & 2 \\
-1 & 2 & 2
\end{pmatrix}
\]

To find the eigenvalues and eigenvectors of this matrix, we follow these steps:

### Step 1: Find the Eigenvalues
The eigenvalues \( \lambda \) are found by solving the characteristic equation:

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

where \( I \) is the identity matrix of the same dimension as \( A \), and \( \det \) denotes the determinant. Let me calculate this.

### Step 2: Find the Eigenvectors
Once we find the eigenvalues \( \lambda \), the corresponding eigenvectors \( v \) are found by solving the equation:

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

Let's compute both the eigenvalues and the eigenvectors.The eigenvalues and eigenvectors of the matrix \( A \) are as follows:

### Eigenvalues:
\[
\lambda_1 = -0.192582404, \quad \lambda_2 = 6.883 \times 10^{-15} \ (\approx 0), \quad \lambda_3 = 5.19258240
\]

### Corresponding Eigenvectors:
1. Eigenvector corresponding to \( \lambda_1 = -0.192582404 \):
   \[
   v_1 = \begin{pmatrix} -0.1228 \\ -0.7605 \\ 0.6377 \end{pmatrix}
   \]

2. Eigenvector corresponding to \( \lambda_2 \approx 0 \):
   \[
   v_2 = \begin{pmatrix} -0.1849 \\ -0.7396 \\ 0.6472 \end{pmatrix}
   \]

3. Eigenvector corresponding to \( \lambda_3 = 5.19258240 \):
   \[
   v_3 = \begin{pmatrix} 0.7695 \\ 0.6213 \\ 0.1482 \end{pmatrix}
   \]

Would you like a detailed explanation of how these were computed or have any questions?

Here are 5 related questions you might find useful:
1. How do we interpret the geometric meaning of eigenvalues and eigenvectors?
2. How can we use eigenvectors to diagonalize a matrix?
3. What is the significance of having an eigenvalue close to zero?
4. What are the applications of eigenvalues in physics and engineering?
5. How are eigenvalues and eigenvectors used in machine learning algorithms?

**Tip:** Eigenvalues can help determine the stability of a system in differential equations.

This page provides a step-by-step solution to finding the eigenvalues and eigenvectors of the 3x3 matrix 2 3 4, 3 1 2, -1 2 2. The solution covers key linear algebra concepts such as the characteristic equation, eigenvalue theorem, and diagonalization. Suitable for undergraduate-level math students.

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