where B = P-¹AP. STEP 1: First note that A 3 00 0 -2 0 0 0 1 P-¹AP = = || |B| = 13 11 13([ STEP 3: Does |B| = O Yes O No -- [³ 1 -2 0 -1 12 0-1 1 1-2 0 -1 12 0 -1 1 000 |A|? 13 10 -20 -9 -6 3 and B are similar. 3 ][ 14 -5 3 00 0-2 0 0 01 STEP 2: Take the determinant of B by expanding along the first row. ↓ 1 Q 11 3 2 -4 1 1 -2 1 1 -1 BEE 1 10(3) - 32-4 11-2 1 1 -1 || -9 14 - 10-3 20 3-5 1-2 0 2 0 -1 1 -1 3
where B = P-¹AP. STEP 1: First note that A 3 00 0 -2 0 0 0 1 P-¹AP = = || |B| = 13 11 13([ STEP 3: Does |B| = O Yes O No -- [³ 1 -2 0 -1 12 0-1 1 1-2 0 -1 12 0 -1 1 000 |A|? 13 10 -20 -9 -6 3 and B are similar. 3 ][ 14 -5 3 00 0-2 0 0 01 STEP 2: Take the determinant of B by expanding along the first row. ↓ 1 Q 11 3 2 -4 1 1 -2 1 1 -1 BEE 1 10(3) - 32-4 11-2 1 1 -1 || -9 14 - 10-3 20 3-5 1-2 0 2 0 -1 1 -1 3
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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![**Educational Website Transcription**
---
### Matrix Similarity and Determinants
**Given Matrices:**
\[ A = \begin{bmatrix} 3 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 13 & 10 & -20 \\ -9 & -6 & 14 \\ 3 & 3 & -5 \end{bmatrix}, \quad P = \begin{bmatrix} 3 & 2 & -4 \\ 1 & 1 & -2 \\ 1 & -1 & 1 \end{bmatrix} \]
\[ P^{-1} = \begin{bmatrix} 1 & -2 & 0 \\ -1 & 1 & 2 \\ 0 & -1 & 1 \end{bmatrix} \]
where \( B = P^{-1}AP \).
---
**STEP 1: Similarity**
**Goal:** Confirm that \( A \) and \( B \) are similar matrices by calculating \( P^{-1}AP \).
\[ P^{-1}AP = \begin{bmatrix} 1 & -2 & 0 \\ -1 & 1 & 2 \\ 0 & -1 & 1 \end{bmatrix} \begin{bmatrix} 3 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 3 & 2 & -4 \\ 1 & 1 & -2 \\ 1 & -1 & 1 \end{bmatrix} \]
**Step-by-step Matrix Multiplication:**
- Multiply the first two matrices:
\[
\begin{bmatrix}
1 & -2 & 0 \\
-1 & 1 & 2 \\
0 & -1 & 1
\end{bmatrix}
\begin{bmatrix}
3 & 0 & 0 \\
0 & -2 & 0 \\
0 & 0 & 1
\end{bmatrix}
=
\begin{bmatrix}
3 & 4 & 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4b64424-2f1e-400b-a2f3-ccc24d7d74e8%2Fbfb72d4b-c61a-4e4e-a9fa-e5540a555d60%2Fwkp5tp_processed.png&w=3840&q=75)
Transcribed Image Text:**Educational Website Transcription**
---
### Matrix Similarity and Determinants
**Given Matrices:**
\[ A = \begin{bmatrix} 3 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 13 & 10 & -20 \\ -9 & -6 & 14 \\ 3 & 3 & -5 \end{bmatrix}, \quad P = \begin{bmatrix} 3 & 2 & -4 \\ 1 & 1 & -2 \\ 1 & -1 & 1 \end{bmatrix} \]
\[ P^{-1} = \begin{bmatrix} 1 & -2 & 0 \\ -1 & 1 & 2 \\ 0 & -1 & 1 \end{bmatrix} \]
where \( B = P^{-1}AP \).
---
**STEP 1: Similarity**
**Goal:** Confirm that \( A \) and \( B \) are similar matrices by calculating \( P^{-1}AP \).
\[ P^{-1}AP = \begin{bmatrix} 1 & -2 & 0 \\ -1 & 1 & 2 \\ 0 & -1 & 1 \end{bmatrix} \begin{bmatrix} 3 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 3 & 2 & -4 \\ 1 & 1 & -2 \\ 1 & -1 & 1 \end{bmatrix} \]
**Step-by-step Matrix Multiplication:**
- Multiply the first two matrices:
\[
\begin{bmatrix}
1 & -2 & 0 \\
-1 & 1 & 2 \\
0 & -1 & 1
\end{bmatrix}
\begin{bmatrix}
3 & 0 & 0 \\
0 & -2 & 0 \\
0 & 0 & 1
\end{bmatrix}
=
\begin{bmatrix}
3 & 4 & 0
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