Determine whether A is diagonalizable. 40 0 8 04 0 0 00-4 0 Loo 0 -4 A = O Yes O No Find an invertible matrix P and a diagonal matrix D such that P-¹AP = D. (Enter each matrix in the form [[row 1], [row 2], ...], where each row is a comma-separated list. If A is not diagonalizable, enter NO SOLUTION.) (D, P) =
Determine whether A is diagonalizable. 40 0 8 04 0 0 00-4 0 Loo 0 -4 A = O Yes O No Find an invertible matrix P and a diagonal matrix D such that P-¹AP = D. (Enter each matrix in the form [[row 1], [row 2], ...], where each row is a comma-separated list. If A is not diagonalizable, enter NO SOLUTION.) (D, P) =
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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Question
![**Determine whether A is diagonalizable.**
A =
\[
\begin{bmatrix}
4 & 0 & 0 & 8 \\
0 & 4 & 0 & 0 \\
0 & 0 & -4 & 0 \\
0 & 0 & 0 & -4 \\
\end{bmatrix}
\]
- [ ] Yes
- [ ] No
**Find an invertible matrix \( P \) and a diagonal matrix \( D \) such that \( P^{-1}AP = D \).**
- Enter each matrix in the form \([[\text{row 1}], [\text{row 2}], \ldots ]\), where each row is a comma-separated list.
- If \( A \) is not diagonalizable, enter NO SOLUTION.
\((D, P) =\) \([]\)
---
**Explanation**
The given matrix \( A \) is a 4x4 matrix. The task is to determine if this matrix is diagonalizable, which means it can be expressed in the form \( P^{-1}AP = D \), where \( D \) is a diagonal matrix and \( P \) is an invertible matrix.
To find if \( A \) is diagonalizable, we need to check the eigenvalues and corresponding eigenvectors. The matrix is diagonalizable if it has a complete set of linearly independent eigenvectors (i.e., its algebraic multiplicity equals its geometric multiplicity for each eigenvalue).
If diagonalizable, fill in matrices \( D \) and \( P \) in the answer box. If not diagonalizable, simply write "NO SOLUTION".](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c4c6df5-8b96-455e-b2de-04bdb043521a%2F83dfb086-66ef-4830-bf7e-15601b329b02%2Fd2km9ua_processed.png&w=3840&q=75)
Transcribed Image Text:**Determine whether A is diagonalizable.**
A =
\[
\begin{bmatrix}
4 & 0 & 0 & 8 \\
0 & 4 & 0 & 0 \\
0 & 0 & -4 & 0 \\
0 & 0 & 0 & -4 \\
\end{bmatrix}
\]
- [ ] Yes
- [ ] No
**Find an invertible matrix \( P \) and a diagonal matrix \( D \) such that \( P^{-1}AP = D \).**
- Enter each matrix in the form \([[\text{row 1}], [\text{row 2}], \ldots ]\), where each row is a comma-separated list.
- If \( A \) is not diagonalizable, enter NO SOLUTION.
\((D, P) =\) \([]\)
---
**Explanation**
The given matrix \( A \) is a 4x4 matrix. The task is to determine if this matrix is diagonalizable, which means it can be expressed in the form \( P^{-1}AP = D \), where \( D \) is a diagonal matrix and \( P \) is an invertible matrix.
To find if \( A \) is diagonalizable, we need to check the eigenvalues and corresponding eigenvectors. The matrix is diagonalizable if it has a complete set of linearly independent eigenvectors (i.e., its algebraic multiplicity equals its geometric multiplicity for each eigenvalue).
If diagonalizable, fill in matrices \( D \) and \( P \) in the answer box. If not diagonalizable, simply write "NO SOLUTION".
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