For the matrix A, find (if possible) a nonsingular matrix P such that P-¹AP is diagonal. (If not possible, enter IMPOSSIBLE.) 1 44] A = -1 P = 1 -1 P-¹AP = 1 1 -3 -3 Verify that P-¹AP is a diagonal matrix with the eigenvalues on the main diagonal. ↓ 1 -2 ← -1
For the matrix A, find (if possible) a nonsingular matrix P such that P-¹AP is diagonal. (If not possible, enter IMPOSSIBLE.) 1 44] A = -1 P = 1 -1 P-¹AP = 1 1 -3 -3 Verify that P-¹AP is a diagonal matrix with the eigenvalues on the main diagonal. ↓ 1 -2 ← -1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Just the last part that is incorrect. Please write out or type in a way that it is clear what the matrix is.
![For the matrix \( A \), find (if possible) a nonsingular matrix \( P \) such that \( P^{-1}AP \) is diagonal. (If not possible, enter IMPOSSIBLE.)
\[
A =
\begin{bmatrix}
1 & \frac{1}{2} \\
- \frac{3}{2} & -1
\end{bmatrix}
\]
\[
P =
\begin{bmatrix}
1 & 1 \\
-1 & -3
\end{bmatrix}
\]
✔️
Verify that \( P^{-1}AP \) is a diagonal matrix with the eigenvalues on the main diagonal.
\[
P^{-1}AP =
\begin{bmatrix}
1 & -2 \\
-3 & -1
\end{bmatrix}
\]
❌
### Explanation:
1. **Matrix \( A \):** This is the original matrix given that needs to be diagonalized if possible.
2. **Matrix \( P \):** A nonsingular matrix which is supposed to diagonalize \( A \). The arrows indicate that \( P \) has been identified as a potential matrix for diagonalization.
3. **Verification of \( P^{-1}AP \):** The result of calculating \( P^{-1}AP \) does not lead to a diagonal matrix, as indicated by the crossed-out mark (❌). The elements on the main diagonal are not eigenvalues of the original matrix, thus the attempt was unsuccessful.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61ff295b-afd4-43c3-8ed2-2cc17b9c2249%2F26a65951-ed31-4191-84bd-703d52839e49%2F8qjlnxm_processed.png&w=3840&q=75)
Transcribed Image Text:For the matrix \( A \), find (if possible) a nonsingular matrix \( P \) such that \( P^{-1}AP \) is diagonal. (If not possible, enter IMPOSSIBLE.)
\[
A =
\begin{bmatrix}
1 & \frac{1}{2} \\
- \frac{3}{2} & -1
\end{bmatrix}
\]
\[
P =
\begin{bmatrix}
1 & 1 \\
-1 & -3
\end{bmatrix}
\]
✔️
Verify that \( P^{-1}AP \) is a diagonal matrix with the eigenvalues on the main diagonal.
\[
P^{-1}AP =
\begin{bmatrix}
1 & -2 \\
-3 & -1
\end{bmatrix}
\]
❌
### Explanation:
1. **Matrix \( A \):** This is the original matrix given that needs to be diagonalized if possible.
2. **Matrix \( P \):** A nonsingular matrix which is supposed to diagonalize \( A \). The arrows indicate that \( P \) has been identified as a potential matrix for diagonalization.
3. **Verification of \( P^{-1}AP \):** The result of calculating \( P^{-1}AP \) does not lead to a diagonal matrix, as indicated by the crossed-out mark (❌). The elements on the main diagonal are not eigenvalues of the original matrix, thus the attempt was unsuccessful.
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