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The image you've provided shows a matrix \( A \) and the corresponding eigenvalues and eigenvectors.

Given:
\[
A = \begin{pmatrix}
3 & 10 & 5 \\
-2 & -3 & -4 \\
3 & 5 & 7
\end{pmatrix}
\]
The eigenvalues are \( \lambda = 3, 2, 2 \).

The corresponding eigenvectors provided are:
\[
\text{Eigenvectors} = \begin{pmatrix} 1 \\ 1 \\ -2 \end{pmatrix}, \begin{pmatrix} 5 \\ 2 \\ -5 \end{pmatrix}
\]

Would you like a detailed explanation on how to calculate the eigenvalues and eigenvectors, or do you have any specific questions related to this problem?

### Related Questions:
1. How do you calculate the eigenvalues of a 3x3 matrix?
2. What is the significance of repeated eigenvalues?
3. How do you determine eigenvectors once you have the eigenvalues?
4. Why might some eigenvalues correspond to multiple eigenvectors?
5. Can two different eigenvectors correspond to the same eigenvalue?

### Tip:
When calculating eigenvectors, always double-check your solutions by substituting them back into the original matrix equation \( A\mathbf{v} = \lambda\mathbf{v} \) to ensure they satisfy the equation.

Find the eigenvalues and eigenvectors for the matrix A given as:
A = [3, 10, 5; -2, -3, -4; 3, 5, 7]

Calculate the eigenvalues and eigenvectors for the matrix A = [3, 10, 5; -2, -3, -4; 3, 5, 7]. This problem explores key linear algebra concepts including the Eigenvalue Theorem and the use of the characteristic equation. Suitable for undergraduate students.

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