3 2 4
2 0 2
4 2 3 Find eigen values and vectors

To find the eigenvalues and eigenvectors of a given matrix, we 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

The eigenvalues \( \lambda \) are solutions to the **characteristic equation**:

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

Where \( I \) is the identity matrix and \( \det \) represents the determinant. Let's calculate the eigenvalues by solving this determinant.

### Step 2: Find the Eigenvectors

Once we have the eigenvalues \( \lambda \), we will find the corresponding eigenvectors by solving:

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

Where \( \mathbf{v} \) is the eigenvector corresponding to eigenvalue \( \lambda \).

Let me calculate that for you.The eigenvalues of the matrix \( A \) are:

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

The corresponding eigenvectors are:

\[
\mathbf{v}_1 = \begin{pmatrix} -0.7454 \\ 0.2981 \\ 0.5963 \end{pmatrix}, \quad
\mathbf{v}_2 = \begin{pmatrix} 0.6667 \\ 0.3333 \\ 0.6667 \end{pmatrix}, \quad
\mathbf{v}_3 = \begin{pmatrix} -0.2076 \\ -0.7760 \\ 0.5956 \end{pmatrix}
\]

Would you like further details on how to calculate the eigenvalues and eigenvectors, or an explanation of how to interpret these results?

Here are five related questions:
1. How do we compute the characteristic polynomial of a matrix?
2. What is the geometric interpretation of eigenvalues and eigenvectors?
3. How are eigenvalues used in diagonalizing a matrix?
4. Can all matrices be diagonalized using their eigenvalues?
5. How do degenerate (repeated) eigenvalues affect eigenvectors?

**Tip:** Eigenvalues and eigenvectors are crucial in many applications, including stability analysis, quantum mechanics, and principal component analysis (PCA).

3 2 4
2 0 2
4 2 3 Find eigenvalues and vectors

Discover how to calculate the eigenvalues and eigenvectors of the 3x3 matrix A = [3, 2, 4; 2, 0, 2; 4, 2, 3]. This detailed guide covers key linear algebra concepts, including the characteristic equation and determinant calculation, along with eigenvector determination. Suitable for advanced high school or undergraduate students.

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