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

The matrix \( A \) is:
\[
A = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 0 & 0 & 1 \end{bmatrix}
\]

### Step 1: Find the Eigenvalues
To find the eigenvalues, we solve the characteristic equation given by:
\[
\det(A - \lambda I) = 0
\]
where \( I \) is the identity matrix, and \( \lambda \) represents the eigenvalues.

So, \( A - \lambda I \) is:
\[
A - \lambda I = \begin{bmatrix} 2 - \lambda & 1 & 1 \\ 1 & 2 - \lambda & 1 \\ 0 & 0 & 1 - \lambda \end{bmatrix}
\]

The determinant of \( A - \lambda I \) is:
\[
\det(A - \lambda I) = (2 - \lambda) \left((2 - \lambda)(1 - \lambda) - (1)(0)\right) - (1) \left(1 \cdot (1 - \lambda) - 1 \cdot 0\right)
\]

Simplifying the determinant:
\[
= (2 - \lambda) \left((2 - \lambda)(1 - \lambda)\right) - (1) \left(1 - \lambda\right)
\]
\[
= (2 - \lambda) \left(2 - 3\lambda + \lambda^2\right) - (1 - \lambda)
\]
\[
= 2(2 - 3\lambda + \lambda^2) - \lambda(2 - 3\lambda + \lambda^2) - (1 - \lambda)
\]

Let me calculate the roots of this determinant equation for eigenvalues.The eigenvalues of the matrix \( A \) are:

\[
\lambda_1 = 1 \quad \text{(with multiplicity 2)} \quad \text{and} \quad \lambda_2 = 3
\]

### Step 2: Find the Eigenvectors
Now, let's find the corresponding eigenvectors for each eigenvalue.

#### Eigenvector for \(\lambda = 1\):

We solve the system \((A - \lambda I) \mathbf{v} = 0\).

Let's calculate the eigenvectors for \(\lambda = 1\).It seems like I can’t do more advanced data analysis right now. Please try again later.

If you would like, I can guide you through finding the eigenvectors manually or assist with other questions! Here’s how you would proceed to find the eigenvectors for \(\lambda = 1\):

1. Substitute \(\lambda = 1\) into \(A - \lambda I\) and solve \((A - I) \mathbf{v} = 0\).
   
If you need more help with this or have another request, please let me know!

### Follow-up Questions:
1. Do you want a step-by-step guide on solving for eigenvectors manually?
2. Would you like to explore eigenvalues and eigenvectors of different matrices?
3. Are you interested in learning about the geometric interpretation of eigenvectors?
4. Do you need help understanding how eigenvectors relate to matrix transformations?
5. Would you like to learn about applications of eigenvalues and eigenvectors in various fields?

**Tip:** Always check the multiplicity of eigenvalues, as it affects the number of independent eigenvectors you need to find!

Find the Eigenvalues and Eigenvectors of the matrix A = [[2, 1, 1], [1, 2, 1], [0, 0, 1]].

Find the eigenvalues and eigenvectors of a 3x3 matrix A = [[2, 1, 1], [1, 2, 1], [0, 0, 1]] with step-by-step instructions. This solution covers key linear algebra concepts such as matrix determinants and eigenvalue calculations, suitable for undergraduate students.

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