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

We are given a 3x3 matrix \( A \):

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

### Step 1: Finding Eigenvalues
The eigenvalues \( \lambda \) of a matrix are the roots of the **characteristic equation**:
\[
\det(A - \lambda I) = 0
\]
where \( I \) is the identity matrix of the same size as \( A \).

We compute the determinant of \( A - \lambda I \), which looks like this:

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

Now, we'll compute the determinant of this matrix and solve for \( \lambda \).

### Step 2: Eigenvectors
Once we have the eigenvalues, we can find the corresponding eigenvectors by solving the system:
\[
(A - \lambda I) \mathbf{v} = 0
\]
for each eigenvalue \( \lambda \), where \( \mathbf{v} \) is the eigenvector associated with \( \lambda \).

I'll begin with calculating the determinant and solving for the eigenvalues and eigenvectors.### Eigenvalues:
The eigenvalues of the matrix \( A \) are:
\[
\lambda_1 = -2.5616, \quad \lambda_2 = 2, \quad \lambda_3 = 1.5616
\]

### Eigenvectors:
The corresponding eigenvectors (each column corresponds to an eigenvalue) are:
\[
\mathbf{v}_1 = \begin{bmatrix} -0.3418 \\ -0.8754 \\ 0.3418 \end{bmatrix}, \quad
\mathbf{v}_2 = \begin{bmatrix} 0.5421 \\ -0.6776 \\ -0.4969 \end{bmatrix}, \quad
\mathbf{v}_3 = \begin{bmatrix} 0.4747 \\ -0.7412 \\ -0.4747 \end{bmatrix}
\]

Would you like further details on how to interpret these results or any additional clarifications?

Here are 5 related questions to deepen your understanding:
1. What is the geometric significance of eigenvectors and eigenvalues?
2. How do the eigenvalues of a matrix relate to its determinant and trace?
3. How can we use eigenvectors to diagonalize a matrix?
4. What is the characteristic polynomial of this matrix?
5. How do eigenvalues affect the stability of a dynamical system?

**Tip:** Eigenvalues can indicate whether a matrix represents a transformation that is scaling, rotating, or shearing vectors in a vector space.

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

Learn how to find the eigenvalues and eigenvectors of a 3x3 matrix. This solution demonstrates key linear algebra concepts, including solving the characteristic equation and finding eigenvectors. Suitable for undergraduate students studying linear algebra.

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