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To determine if the matrix \( A \) is diagonalizable, we'll need to check if the algebraic multiplicity of each eigenvalue matches its geometric multiplicity.

The matrix given is:
\[ A = \begin{bmatrix} -1 & 2 & -2 \\ 1 & 2 & 1 \\ -1 & -1 & 0 \end{bmatrix} \]

Let's start by finding the eigenvalues of \( A \). The eigenvalues are the solutions to the characteristic polynomial equation \( \det(A - \lambda I) = 0 \), where \( I \) is the identity matrix and \( \lambda \) are the eigenvalues.

We'll calculate the characteristic polynomial next.

1. First, construct the matrix \( A - \lambda I \):
\[ A - \lambda I = \begin{bmatrix} -1-\lambda & 2 & -2 \\ 1 & 2-\lambda & 1 \\ -1 & -1 & -\lambda \end{bmatrix} \]

2. Compute the determinant of \( A - \lambda I \):
\[ \text{det}(A - \lambda I) = (-1-\lambda)((2-\lambda)(-\lambda) + 1) - 2(1 \cdot (-\lambda) + 1) + (-2)(-1 + (2-\lambda)) \]

Let's perform these computations and solve for \( \lambda \).The eigenvalues of the matrix \( A \) are \( \lambda = 1 \), \( \lambda = -\sqrt{5} \), and \( \lambda = \sqrt{5} \).

Next, we'll find the eigenvectors for each eigenvalue to determine the geometric multiplicity. If the geometric multiplicity of each eigenvalue matches its algebraic multiplicity (which is 1 for each eigenvalue since they are all distinct), then the matrix is diagonalizable.

Let's start by finding the eigenvectors for each eigenvalue:

1. For \( \lambda = 1 \)
2. For \( \lambda = -\sqrt{5} \)
3. For \( \lambda = \sqrt{5} \)

I'll compute the eigenvectors for \( \lambda = 1 \) first.For \( \lambda = 1 \), the eigenvector is:
\[
\begin{bmatrix}
-1 \\
0 \\
1
\end{bmatrix}
\]
This implies that the geometric multiplicity of \( \lambda = 1 \) is 1, which matches its algebraic multiplicity.

Next, we'll find the eigenvectors for \( \lambda = -\sqrt{5} \) and \( \lambda = \sqrt{5} \). Let's proceed with \( \lambda = -\sqrt{5} \).It appears I forgot to import the `sqrt` function. Let me fix that and recalculate the eigenvectors for \( \lambda = -\sqrt{5} \).It seems like I can’t do more advanced data analysis right now. Please try again later.

Do you need help with anything else or have any other questions related to this topic? Here are some related questions you might consider:

1. How do you calculate the characteristic polynomial of a matrix?
2. What is the geometric multiplicity and how does it relate to diagonalization?
3. How can eigenvectors be used to diagonalize a matrix?
4. What are the implications of having distinct eigenvalues for a matrix?
5. Can a matrix have more eigenvectors than its rank?

Tip: When checking if a matrix is diagonalizable, always ensure you check both the algebraic and geometric multiplicities of each eigenvalue.

Reduce the matrix A = [[-1, 2, -2], [1, 2, 1], [-1, -1, 0]] into diagonalizable form?

Learn how to diagonalize the matrix A = [[-1, 2, -2], [1, 2, 1], [-1, -1, 0]] by finding its eigenvalues and eigenvectors. This linear algebra problem covers key concepts in matrix diagonalization, including the characteristic polynomial and eigenvalue equations. Suitable for undergraduate students.

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