To solve this problem, we need to use the information about the matrix $A$'s eigenvalues and corresponding eigenvectors to construct matrices $P$ and $D$ such that $A = PDP^{-1}$, where:

1. Matrix $D$: This is a diagonal matrix containing the eigenvalues of $A$.
2. Matrix $P$: This matrix has the eigenvectors of $A$ as its columns.

Step-by-Step Solution:

1. Identify $D$: The diagonal matrix $D$ will contain the eigenvalues of $A$. We are given:

   • Eigenvalue $3$ with its corresponding eigenvector $\begin{bmatrix} -1 \\ 2 \end{bmatrix}$.
   • Eigenvalue $-2$ with its corresponding eigenvector $\begin{bmatrix} -1 \\ 1 \end{bmatrix}$.

   So, $D = \begin{bmatrix} 3 & 0 \\ 0 & -2 \end{bmatrix}$.

2. Construct $P$: The matrix $P$ will have the eigenvectors as columns. Therefore, $P = \begin{bmatrix} -1 & -1 \\ 2 & 1 \end{bmatrix}.$

3. Find $P^{-1}$: To complete the decomposition, we need the inverse of $P$, $P^{-1}$.

   • $P = \begin{bmatrix} -1 & -1 \\ 2 & 1 \end{bmatrix}$.
   • To find $P^{-1}$, we use the formula for the inverse of a 2x2 matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$: $P^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}.$
   • For $P = \begin{bmatrix} -1 & -1 \\ 2 & 1 \end{bmatrix}$:
     • Determinant $\det(P) = (-1)(1) - (-1)(2) = -1 + 2 = 1$.
     • Thus, $P^{-1} = \begin{bmatrix} 1 & 1 \\ -2 & -1 \end{bmatrix}.$

4. Express $A = PDP^{-1}$: Now, we have $A = \begin{bmatrix} -1 & -1 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} 3 & 0 \\ 0 & -2 \end{bmatrix} \begin{bmatrix} 1 & 1 \\ -2 & -1 \end{bmatrix}.$

Summary of the Answer:

• $P = \begin{bmatrix} -1 & -1 \\ 2 & 1 \end{bmatrix}$
• $D = \begin{bmatrix} 3 & 0 \\ 0 & -2 \end{bmatrix}$
• $P^{-1} = \begin{bmatrix} 1 & 1 \\ -2 & -1 \end{bmatrix}$

Would you like further details on any step or explanation on how to multiply these matrices if needed?

Related Questions:

1. How do you find the inverse of a 3x3 matrix?
2. What are the properties of diagonalizable matrices?
3. How does the eigenvalue decomposition apply to larger matrices?
4. What conditions must a matrix meet to be diagonalizable?
5. How can you verify if a matrix is symmetric or orthogonal?

Tip:

Diagonalization is only possible if a matrix has enough linearly independent eigenvectors to form the matrix $P$.