Find the eigen values and eigen vector of the following matrix 
A = 3 2 4
       2 0 2
       4 2 3

To find the eigenvalues and eigenvectors of matrix \( A \), follow these steps:

Given matrix \( A \):

\[
A = \begin{pmatrix} 
3 & 2 & 4 \\
2 & 0 & 2 \\
4 & 2 & 3 
\end{pmatrix}
\]

### Step 1: Find the eigenvalues
Eigenvalues are obtained by solving the characteristic equation:

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

Where:
- \( A \) is the matrix,
- \( \lambda \) is the eigenvalue,
- \( I \) is the identity matrix of the same size as \( A \).

The matrix \( A - \lambda I \) is:

\[
A - \lambda I = \begin{pmatrix} 
3 - \lambda & 2 & 4 \\
2 & 0 - \lambda & 2 \\
4 & 2 & 3 - \lambda 
\end{pmatrix}
\]

Now, compute the determinant of this matrix:

\[
\text{det}(A - \lambda I) = \begin{vmatrix} 
3 - \lambda & 2 & 4 \\
2 & -\lambda & 2 \\
4 & 2 & 3 - \lambda 
\end{vmatrix}
\]

Expanding this determinant will give the characteristic polynomial, and solving this for \( \lambda \) will yield the eigenvalues.

### Step 2: Find the eigenvectors
Once the eigenvalues are determined, the eigenvectors can be found by solving the system:

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

For each eigenvalue \( \lambda \), we substitute it back into \( A - \lambda I \) and solve for the vector \( \vec{v} \) (the eigenvector) that satisfies this equation.

I'll now compute the determinant to find the eigenvalues and then determine the eigenvectors.The eigenvalues of the matrix \( A \) are:

\[
\lambda_1 = -1, \quad \lambda_2 = 8, \quad \lambda_3 = -1
\]

The corresponding eigenvectors are:

\[
\vec{v_1} = \begin{pmatrix} -0.7454 \\ 0.2981 \\ 0.5963 \end{pmatrix}, \quad
\vec{v_2} = \begin{pmatrix} 0.6667 \\ 0.3333 \\ 0.6667 \end{pmatrix}, \quad
\vec{v_3} = \begin{pmatrix} -0.2076 \\ -0.7760 \\ 0.5956 \end{pmatrix}
\]

### Summary:
- Eigenvalues: \( -1, 8, -1 \)
- Eigenvectors: Listed above.

Would you like a detailed step-by-step breakdown of the calculations?

Here are 5 follow-up questions related to eigenvalues and eigenvectors:
1. How do the eigenvectors relate to the original matrix transformation?
2. What is the geometric significance of the eigenvalues?
3. How do you normalize an eigenvector?
4. What is the characteristic equation for this matrix?
5. Can a matrix have complex eigenvalues?

**Tip**: Eigenvalues tell you the factor by which the transformation scales a vector, while eigenvectors show the directions along which this scaling occurs.

Find the eigenvalues and eigenvectors of the following matrix A = [3, 2, 4; 2, 0, 2; 4, 2, 3].

Learn how to calculate the eigenvalues and eigenvectors of the 3x3 matrix A = [3, 2, 4; 2, 0, 2; 4, 2, 3]. This detailed solution covers key linear algebra concepts, including the characteristic equation and eigenvector calculation. Suitable for Grades 11-12 and undergraduate students.

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