Find the Eigenvalues and Eigenvectors of the matrix A =              
 −2 2 3
  2 1 −6
−1 −2 0

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

Given the matrix:

\[
A = \begin{pmatrix} 
-2 & 2 & 3 \\ 
2 & 1 & -6 \\ 
-1 & -2 & 0 
\end{pmatrix}
\]

### Step 1: Find the Eigenvalues
The eigenvalues of a matrix are found by solving the characteristic equation:

\[
\det(A - \lambda I) = 0
\]

where \( \lambda \) is the eigenvalue, and \( I \) is the identity matrix of the same size as \( A \). First, compute \( A - \lambda I \):

\[
A - \lambda I = \begin{pmatrix}
-2-\lambda & 2 & 3 \\
2 & 1-\lambda & -6 \\
-1 & -2 & -\lambda
\end{pmatrix}
\]

Next, compute the determinant:

\[
\det(A - \lambda I) = \det \begin{pmatrix}
-2-\lambda & 2 & 3 \\
2 & 1-\lambda & -6 \\
-1 & -2 & -\lambda
\end{pmatrix}
\]

Now, let's calculate this determinant.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like a detailed walkthrough on how to compute the determinant manually, or any other assistance with this? If you have any questions, feel free to ask!

Here are 5 follow-up questions related to eigenvalues and eigenvectors:
1. How are eigenvalues used in practical applications like physics or engineering?
2. Can every square matrix have real eigenvalues?
3. What is the geometric interpretation of eigenvectors?
4. How do you determine if a matrix is diagonalizable?
5. How are eigenvalues connected to the trace and determinant of a matrix?

**Tip**: When solving for eigenvalues, remember that solving the characteristic equation often leads to polynomial equations, which can have complex roots!

Find the Eigenvalues and Eigenvectors of the matrix A = [-2 2 3; 2 1 -6; -1 -2 0]

This guide explains how to find the eigenvalues and eigenvectors of the matrix A = [-2 2 3; 2 1 -6; -1 -2 0]. It includes detailed steps for constructing the characteristic equation and calculating the determinant, essential concepts in linear algebra. Suitable for undergraduate students.

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