Diagonalize the matrix A = [[2, 2, 0], [2, 1, 1], [- 7, 2, - 3]] find eigen vectors 



To diagonalize a matrix \( A \), we need to:

1. Find the eigenvalues of \( A \) by solving the characteristic equation \( \det(A - \lambda I) = 0 \), where \( \lambda \) is the eigenvalue and \( I \) is the identity matrix.
2. Find the corresponding eigenvectors for each eigenvalue.

The matrix \( A \) is given as:

\[
A = \begin{pmatrix} 2 & 2 & 0 \\ 2 & 1 & 1 \\ -7 & 2 & -3 \end{pmatrix}
\]

### Step 1: Find Eigenvalues
The characteristic equation is \( \det(A - \lambda I) = 0 \), where \( I \) is the identity matrix.

Let's calculate this determinant first:

\[
\det \left( \begin{pmatrix} 2 & 2 & 0 \\ 2 & 1 & 1 \\ -7 & 2 & -3 \end{pmatrix} - \lambda \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \right) = 0
\]

This results in:

\[
\det \begin{pmatrix} 2 - \lambda & 2 & 0 \\ 2 & 1 - \lambda & 1 \\ -7 & 2 & -3 - \lambda \end{pmatrix} = 0
\]

Now, let me calculate the determinant and find the eigenvalues.The eigenvalues of the matrix \( A \) are:

\[
\lambda_1 = -4, \quad \lambda_2 = 3, \quad \lambda_3 = 1
\]

The corresponding eigenvectors are:

\[
v_1 = \begin{pmatrix} 0.0747 \\ -0.2242 \\ 0.9717 \end{pmatrix}, \quad
v_2 = \begin{pmatrix} -0.6667 \\ -0.3333 \\ 0.6667 \end{pmatrix}, \quad
v_3 = \begin{pmatrix} -0.4364 \\ 0.2182 \\ 0.8729 \end{pmatrix}
\]

### Conclusion:
The matrix \( A \) can be diagonalized as \( A = PDP^{-1} \), where \( D \) is the diagonal matrix with eigenvalues \( \lambda_1, \lambda_2, \lambda_3 \), and the columns of \( P \) are the corresponding eigenvectors.

Would you like more details on the diagonalization process or on any other step?

### Follow-up questions:
1. How do you calculate the determinant of a 3x3 matrix?
2. How can we verify if a given vector is an eigenvector of a matrix?
3. What are the steps for finding \( P^{-1} \) once you have \( P \)?
4. Can the matrix \( A \) have repeated eigenvalues, and what does that imply?
5. How does the matrix's diagonal form help in solving differential equations?

**Tip:** Eigenvectors corresponding to distinct eigenvalues are linearly independent, which ensures the diagonalization of the matrix is possible.

Diagonalize the matrix A = [[2, 2, 0], [2, 1, 1], [-7, 2, -3]] and find eigenvectors.

Learn how to diagonalize the matrix A = [[2, 2, 0], [2, 1, 1], [-7, 2, -3]] by finding its eigenvalues and corresponding eigenvectors. This step-by-step solution covers key linear algebra concepts, including the characteristic equation and matrix diagonalization. Suitable for undergraduate students.

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