The eigen vector of a 3 by 3 matrix A with corresponding to the eigen values 2,3,6 are ( 1 0 -1 )transpose,( 1 1 1 )transpose , ( -1 2 -1 )transpose respectively find the matrix a

To find the matrix \( A \) given its eigenvectors and corresponding eigenvalues, we can use the following approach:

### Given:
- Eigenvalues: \( \lambda_1 = 2 \), \( \lambda_2 = 3 \), \( \lambda_3 = 6 \)
- Corresponding eigenvectors: 
  - \( \mathbf{v}_1 = \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix} \)
  - \( \mathbf{v}_2 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \)
  - \( \mathbf{v}_3 = \begin{pmatrix} -1 \\ 2 \\ -1 \end{pmatrix} \)

### Steps:
1. **Form the matrix of eigenvectors**: This matrix \( P \) is formed by placing the eigenvectors as columns:
   \[
   P = \begin{pmatrix} 
   1 & 1 & -1 \\
   0 & 1 & 2 \\
   -1 & 1 & -1
   \end{pmatrix}
   \]

2. **Form the diagonal matrix of eigenvalues** \( D \):
   \[
   D = \begin{pmatrix} 
   2 & 0 & 0 \\
   0 & 3 & 0 \\
   0 & 0 & 6
   \end{pmatrix}
   \]

3. **Compute the matrix \( A \)** using the relation \( A = PDP^{-1} \).

First, we'll compute the inverse of \( P \) and then multiply \( PDP^{-1} \) to find \( A \).

Let's calculate it.The matrix \( A \) is given by

Learn how to find the matrix A given its eigenvalues and corresponding eigenvectors. This problem involves constructing a matrix from eigenvectors and eigenvalues using the matrix diagonalization formula A = PDP^{-1}. Suitable for advanced level students and professionals in linear algebra.

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