Find the eigenvalues and corresponding vectors of the following matrix, and diagonalise if possible. Find the matrix which diagonolise it.  A= 1 -1 2 2 -2 4 3 -3 6 

We are given the matrix \( A \):

\[
A = \begin{pmatrix}
1 & -1 & 2 \\
2 & -2 & 4 \\
3 & -3 & 6
\end{pmatrix}
\]

To find the eigenvalues and eigenvectors, we need to:

1. Compute the characteristic equation \( \det(A - \lambda I) = 0 \) to find the eigenvalues \( \lambda \).
2. Solve \( (A - \lambda I) \mathbf{v} = 0 \) to find the eigenvectors corresponding to each eigenvalue \( \lambda \).
3. Diagonalize the matrix if it is diagonalizable by finding the matrix \( P \) that diagonalizes \( A \) (composed of the eigenvectors) and the diagonal matrix \( D \) (composed of the eigenvalues).

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Here are 5 related questions:
1. What are the general steps for diagonalizing a matrix?
2. What is the significance of eigenvalues and eigenvectors in matrix transformations?
3. Can all matrices be diagonalized?
4. How do you interpret a matrix that has repeated eigenvalues?
5. What is the relationship between a matrix’s determinant and its eigenvalues?

**Tip**: If the matrix isn't diagonalizable, it may still be decomposable into other forms, like Jordan canonical form.

Find the eigenvalues and corresponding vectors of the following matrix, and diagonalise if possible. Find the matrix which diagonalises it. A = [1 -1 2; 2 -2 4; 3 -3 6]

Learn how to find the eigenvalues and eigenvectors of the matrix A = [1 -1 2; 2 -2 4; 3 -3 6], and determine whether it can be diagonalized. This problem covers key linear algebra concepts and matrix transformations. Suitable for advanced high school or undergraduate students.

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