To find the eigenvector corresponding to the eigenvalue \(\lambda = 1\), we solve the equation:

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

where \( A \) is the given matrix and \( I \) is the identity matrix.

Given:

\[
A = \begin{pmatrix}
2 & 1 & -1 \\
1 & 1 & -2 \\
-1 & -2 & 1
\end{pmatrix}
\]

Substitute \(\lambda = 1\) into the equation:

\[
A - I = \begin{pmatrix}
2-1 & 1 & -1 \\
1 & 1-1 & -2 \\
-1 & -2 & 1-1
\end{pmatrix} = \begin{pmatrix}
1 & 1 & -1 \\
1 & 0 & -2 \\
-1 & -2 & 0
\end{pmatrix}
\]

Now, solve the system:

\[
\begin{pmatrix}
1 & 1 & -1 \\
1 & 0 & -2 \\
-1 & -2 & 0
\end{pmatrix}
\begin{pmatrix}
x \\
y \\
z
\end{pmatrix} = \begin{pmatrix}
0 \\
0 \\
0
\end{pmatrix}
\]

I'll solve this system for \(\mathbf{v}\).The eigenvector corresponding to the eigenvalue \(\lambda = 1\) is:

\[
\mathbf{v} = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}
\]

Would you like more details or have any further questions?

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**Related Questions:**
1. How do eigenvectors differ for distinct eigenvalues?
2. What is the significance of the eigenvector's direction?
3. Can eigenvectors be scaled by any non-zero scalar?
4. How is the eigenvector used in the diagonalization process?
5. What if the matrix \( A \) is symmetric? How do eigenvectors behave?
6. How can eigenvectors help in simplifying matrix functions?
7. What is the physical interpretation of eigenvectors in mechanics?
8. How do eigenvectors relate to principal component analysis?

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

Learn how to find the eigenvector corresponding to the eigenvalue lambda = 1 using a step-by-step approach. This problem involves a matrix A with specific entries and requires solving the characteristic equation. Suitable for advanced undergraduate students studying linear algebra.

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