For the below matrices
i) Find the eigen values and the eigen vectors.
[■(9&-1&9@-1&-1&3@-7&1&-7)]

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

### Given matrix \( A \):

\[
A = \begin{bmatrix}
9 & -1 & 9 \\
-1 & -1 & 3 \\
-7 & 1 & -7
\end{bmatrix}
\]

### Step 1: Eigenvalues
The eigenvalues \( \lambda \) are the solutions of the characteristic equation:

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

Where \( I \) is the identity matrix of the same size as \( A \), and \( \lambda \) is a scalar.

### Step 2: Eigenvectors
Once the eigenvalues are found, we substitute each \( \lambda \) into the equation \( (A - \lambda I)\mathbf{v} = 0 \), where \( \mathbf{v} \) is the eigenvector associated with \( \lambda \).

Let me compute the eigenvalues and eigenvectors for this matrix.The eigenvalues and eigenvectors of the given matrix are:

### Eigenvalues:
\[
\lambda_1 = -2.5616, \quad \lambda_2 = 2, \quad \lambda_3 = 1.5616
\]

### Eigenvectors (corresponding to each eigenvalue):
For \( \lambda_1 = -2.5616 \):
\[
\mathbf{v}_1 = \begin{bmatrix}
-0.3418 \\
-0.8754 \\
0.3418
\end{bmatrix}
\]

For \( \lambda_2 = 2 \):
\[
\mathbf{v}_2 = \begin{bmatrix}
0.5421 \\
-0.6776 \\
-0.4969
\end{bmatrix}
\]

For \( \lambda_3 = 1.5616 \):
\[
\mathbf{v}_3 = \begin{bmatrix}
0.4747 \\
-0.7412 \\
-0.4747
\end{bmatrix}
\]

Would you like more details or have any questions?

Here are 5 questions to further explore this topic:
1. How do you interpret the geometric significance of eigenvalues?
2. How do eigenvectors relate to the transformation represented by a matrix?
3. What happens when a matrix has repeated eigenvalues?
4. How would you calculate the determinant and trace of the matrix using eigenvalues?
5. Can you explain the relation between eigenvalues and matrix diagonalization?

**Tip:** Eigenvalues can provide insights into the stability of a system modeled by a matrix (e.g., in dynamical systems).

For the below matrices i) Find the eigenvalues and the eigenvectors. [■(9&-1&9@-1&-1&3@-7&1&-7)]

This page provides a detailed solution to finding the eigenvalues and eigenvectors of a 3x3 matrix. Learn how to solve the characteristic equation and compute eigenvectors using matrix algebra. Suitable for college or university students studying linear algebra.

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