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Prove that if A and B are similar, then |A| = |B|. Is the converse true? Illustrate the results using the matrices -1 0 0 A = 03 0 B = 00-2 [ 35 4 4 0 -1 -1 1 1 0 4 6 P = 1 1 1 P-1 -1 1 -1 = -5 -5 -7 1 1 2 0-1 1 where B = P-1AP. STEP 1: First note that A and B are similar. 1 1 0 -1 0 0 0 -1 -1 P-1 AP -1 1 -1 03 0 1 1 1 = 0-1 1 00-2 1 1 2 0 0 1 1 0 -1 1-1 0 3 0 = 0-1 1 -2 II × 0 -1 -1 1 1 1 1 1 2 0 ⇓ 1 具食 - 4 5 -5-7 6 54 +4 -5 -5 STEP 2: Take the determinant of B by expanding along the first row. 4 6 |B|: = -5 -7 = 3 2 6 STEP 3: Does |B| Yes No = |A|? 4(-5)+4 -5

Prove that if A and B are similar, then |A| = |B|. Is the converse true? Illustrate the results using the matrices -1 0 0 A = 03 0 B = 00-2 [ 35 4 4 0 -1 -1 1 1 0 4 6 P = 1 1 1 P-1 -1 1 -1 = -5 -5 -7 1 1 2 0-1 1 where B = P-1AP. STEP 1: First note that A and B are similar. 1 1 0 -1 0 0 0 -1 -1 P-1 AP -1 1 -1 03 0 1 1 1 = 0-1 1 00-2 1 1 2 0 0 1 1 0 -1 1-1 0 3 0 = 0-1 1 -2 II × 0 -1 -1 1 1 1 1 1 2 0 ⇓ 1 具食 - 4 5 -5-7 6 54 +4 -5 -5 STEP 2: Take the determinant of B by expanding along the first row. 4 6 |B|: = -5 -7 = 3 2 6 STEP 3: Does |B| Yes No = |A|? 4(-5)+4 -5

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Prove that if A and B are similar, then |A| = |B|. Is the converse true? Illustrate the results using the matrices -1 0 0 A = 03 0 B = 00-2 [ 35 4 4 0 -1 -1 1 1 0 4 6 P = 1 1 1 P-1 -1 1 -1 = -5 -5 -7 1 1 2 0-1 1 where B = P-1AP. STEP 1: First note that A and B are similar. 1 1 0 -1 0 0 0 -1 -1 P-1 AP -1 1 -1 03 0 1 1 1 = 0-1 1 00-2 1 1 2 0 0 1 1 0 -1 1-1 0 3 0 = 0-1 1 -2 II × 0 -1 -1 1 1 1 1 1 2 0 ⇓ 1 具食 - 4 5 -5-7 6 54 +4 -5 -5 STEP 2: Take the determinant of B by expanding along the first row. 4 6 |B|: = -5 -7 = 3 2 6 STEP 3: Does |B| Yes No = |A|? 4(-5)+4 -5
Transcribed Image Text:Prove that if A and B are similar, then |A| = |B|. Is the converse true? Illustrate the results using the matrices -1 0 0 A = 03 0 B = 00-2 [ 35 4 4 0 -1 -1 1 1 0 4 6 P = 1 1 1 P-1 -1 1 -1 = -5 -5 -7 1 1 2 0-1 1 where B = P-1AP. STEP 1: First note that A and B are similar. 1 1 0 -1 0 0 0 -1 -1 P-1 AP -1 1 -1 03 0 1 1 1 = 0-1 1 00-2 1 1 2 0 0 1 1 0 -1 1-1 0 3 0 = 0-1 1 -2 II × 0 -1 -1 1 1 1 1 1 2 0 ⇓ 1 具食 - 4 5 -5-7 6 54 +4 -5 -5 STEP 2: Take the determinant of B by expanding along the first row. 4 6 |B|: = -5 -7 = 3 2 6 STEP 3: Does |B| Yes No = |A|? 4(-5)+4 -5
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