https://file.aiquickdraw.com/mathbot/file/1733801174918k1wzqw8q.jpg

To find matrices \( P \) and \( D \) such that \( A = PDP^{-1} \), we need:
1. The eigenvalues of \( A \).
2. The eigenvectors corresponding to these eigenvalues.

### Finding eigenvalues:

For a triangular matrix, the eigenvalues are the entries on the diagonal. So, the eigenvalues of \( A \) are:
- \( \lambda_1 = 1 \)
- \( \lambda_2 = 4 \)
- \( \lambda_3 = 6 \)

### Finding eigenvectors:

1. **Eigenvector for \(\lambda_1 = 1\):**

   Solve \( (A - I)x = 0 \).
   
   \[
   \begin{pmatrix}
   0 & 2 & 3 \\
   0 & 3 & 5 \\
   0 & 0 & 5
   \end{pmatrix}
   \begin{pmatrix}
   x_1 \\ x_2 \\ x_3
   \end{pmatrix}
   = 
   \begin{pmatrix}
   0 \\ 0 \\ 0
   \end{pmatrix}
   \]

   \( x_3 \) is free. Choose \( x_3 = 1 \), giving \( x_2 = -5/3 \) and \( x_1 = -1/3 \).

   Eigenvector: \(\begin{pmatrix} -1/3 \\ -5/3 \\ 1 \end{pmatrix}\).

2. **Eigenvector for \(\lambda_2 = 4\):**

   Solve \( (A - 4I)x = 0 \).
   
   \[
   \begin{pmatrix}
   -3 & 2 & 3 \\
   0 & 0 & 5 \\
   0 & 0 & 2
   \end{pmatrix}
   \begin{pmatrix}
   x_1 \\ x_2 \\ x_3
   \end{pmatrix}
   = 
   \begin{pmatrix}
   0 \\ 0 \\ 0
   \end{pmatrix}
   \]

   \( x_3 \) is free. Choose \( x_3 = 1 \), giving \( x_2 = 0 \) and \( x_1 = 1 \).

   Eigenvector: \(\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\).

3. **Eigenvector for \(\lambda_3 = 6\):**

   Solve \( (A - 6I)x = 0 \).
   
   \[
   \begin{pmatrix}
   -5 & 2 & 3 \\
   0 & -2 & 5 \\
   0 & 0 & 0
   \end{pmatrix}
   \begin{pmatrix}
   x_1 \\ x_2 \\ x_3
   \end{pmatrix}
   = 
   \begin{pmatrix}
   0 \\ 0 \\ 0
   \end{pmatrix}
   \]

   \( x_2 \) is free. Choose \( x_2 = 1 \), giving \( x_3 = 2/5 \) and \( x_1 = 1/5 \).

   Eigenvector: \(\begin{pmatrix} 1/5 \\ 1 \\ 2/5 \end{pmatrix}\).

### Construct matrices \( D \) and \( P \):

\[
D = \begin{pmatrix}
1 & 0 & 0 \\
0 & 4 & 0 \\
0 & 0 & 6
\end{pmatrix}
\]

\[
P = \begin{pmatrix}
-1/3 & 1 & 1/5 \\
-5/3 & 0 & 1 \\
1 & 0 & 2/5 
\end{pmatrix}
\]

Now \( A = PDP^{-1} \).

Let A be the 3 × 3 matrix given by A = [[1, 2, 3], [0, 4, 5], [0, 0, 6]]. Find an invertible matrix P and a diagonal matrix D such that A = PDP⁻¹.

Explore the step-by-step diagonalization of a 3 × 3 matrix A = [[1, 2, 3], [0, 4, 5], [0, 0, 6]]. Learn to compute eigenvalues, eigenvectors, and construct matrices P and D for A = PDP⁻¹. Suitable for college-level linear algebra studies.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation