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We are tasked with finding the eigenvectors of matrix \( A \), where:

\[
A = \begin{pmatrix} 
1 & 1 & 1 \\
-1 & -3 & -3 \\
2 & 4 & 4
\end{pmatrix}
\]

### Steps to Find the Eigenvectors:

1. **Find the eigenvalues**:
   We start by solving the characteristic equation:
   \[
   \det(A - \lambda I) = 0
   \]
   where \( I \) is the identity matrix and \( \lambda \) represents the eigenvalues.

2. **Find the eigenvectors**:
   For each eigenvalue \( \lambda \), solve the system:
   \[
   (A - \lambda I) \mathbf{v} = 0
   \]
   to find the eigenvector \( \mathbf{v} \).

Let me calculate this step by step.The eigenvalues of matrix \( A \) are approximately:

\[
\lambda_1 = 2, \quad \lambda_2 \approx 0, \quad \lambda_3 \approx 0
\]

The corresponding eigenvectors (each column corresponds to an eigenvector) are:

\[
\mathbf{v}_1 = \begin{pmatrix} 0.2673 \\ -0.5345 \\ 0.8018 \end{pmatrix}, \quad
\mathbf{v}_2 = \begin{pmatrix} -5.86 \times 10^{-9} \\ 0.7071 \\ -0.7071 \end{pmatrix}, \quad
\mathbf{v}_3 = \begin{pmatrix} -5.86 \times 10^{-9} \\ -0.7071 \\ 0.7071 \end{pmatrix}
\]

These are the eigenvectors corresponding to the eigenvalues \( 2 \), \( 0 \), and \( 0 \), respectively.

Would you like more details or further clarifications?

### Related Questions:
1. How do we verify if a vector is an eigenvector of a matrix?
2. What is the physical significance of eigenvalues and eigenvectors?
3. How do eigenvalues help in diagonalizing a matrix?
4. What is the characteristic polynomial of a matrix, and how is it related to eigenvalues?
5. Can a matrix have complex eigenvalues and eigenvectors?

**Tip:** Eigenvectors corresponding to distinct eigenvalues are linearly independent.

Find the eigenvectors for matrix A = [[1, 1, 1], [-1, -3, -3], [2, 4, 4]]

Learn how to find the eigenvectors of matrix A = [[1, 1, 1], [-1, -3, -3], [2, 4, 4]] using the characteristic equation and solving for eigenvalues. This problem involves linear algebra concepts like eigenvalues and eigenvectors, suitable for undergraduate students.

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