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Find eigen valus and eigen vector for a=(7 -2 0 

The given problem is to find the eigenvalues and eigenvectors of the matrix \( A \):

\[
A = \begin{pmatrix}
7 & -2 & 0 \\
-2 & 6 & -2 \\
0 & -2 & 5 
\end{pmatrix}
\]

To find the eigenvalues, we need to solve the characteristic equation, which is derived from \( \det(A - \lambda I) = 0 \), where \( I \) is the identity matrix and \( \lambda \) represents the eigenvalues.

I'll compute the eigenvalues and eigenvectors step-by-step. Let me begin this process.The eigenvalues of the matrix \( A \) are:

\[
\lambda_1 = 9, \quad \lambda_2 = 6, \quad \lambda_3 = 3
\]

The corresponding eigenvectors are:

\[
\text{For } \lambda = 9: \quad \begin{pmatrix} -\frac{2}{3} \\ \frac{2}{3} \\ -\frac{1}{3} \end{pmatrix}
\]

\[
\text{For } \lambda = 6: \quad \begin{pmatrix} -\frac{2}{3} \\ -\frac{1}{3} \\ \frac{2}{3} \end{pmatrix}
\]

\[
\text{For } \lambda = 3: \quad \begin{pmatrix} \frac{1}{3} \\ \frac{2}{3} \\ \frac{2}{3} \end{pmatrix}
\]

These eigenvectors are normalized and correspond to each eigenvalue.

Would you like more details or have any further questions?

Here are 5 related questions:
1. How can we verify if the computed eigenvectors are correct?
2. What is the significance of eigenvalues and eigenvectors in applications?
3. Can you explain how to normalize an eigenvector?
4. How does the characteristic equation relate to eigenvalues?
5. What changes if we apply a transformation using matrix \( A \) to these eigenvectors?

**Tip:** Eigenvectors corresponding to different eigenvalues are always linearly independent.

Find the eigenvalues and eigenvectors of the matrix A = [[7, -2, 0], [-2, 6, -2], [0, -2, 5]]

Learn how to calculate the eigenvalues and eigenvectors of the matrix A = [[7, -2, 0], [-2, 6, -2], [0, -2, 5]]. This problem involves solving the characteristic equation and using linear algebra techniques. Suitable for college-level students studying linear algebra.

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