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: 1 & 1 & -1 \\ 0 & 1 & 2 \\ -1 & 1 & -1 \end{pmatrix}$$
2. Form the diagonal matrix of eigenvalues $D$: 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 6 \end{pmatrix}$$
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