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3. Two n xn matrices A, B are called similar if there exists an invertible n x n matrix S such that A = SBS-1. If A is similar to B, then B is also similar to A, since B = S-1AS. (i) Show that the two 2 x 2 matrices a- (: :). 0 1 0 0 B - (: :) - a 0 0 1 0 %3D are similar. (ii) Consider the two 2 x 2 invertible matrices (: ?) B-(; ) 1 0 A = 1 0 B = 0 1 2 1 Are the matrices similar? (iii) Consider the two 2 × 2 invertible matrices c - (; :). (: :) C = D = 1 Are the matrices similar?

3. Two n xn matrices A, B are called similar if there exists an invertible n x n matrix S such that A = SBS-1. If A is similar to B, then B is also similar to A, since B = S-1AS. (i) Show that the two 2 x 2 matrices a- (: :). 0 1 0 0 B - (: :) - a 0 0 1 0 %3D are similar. (ii) Consider the two 2 x 2 invertible matrices (: ?) B-(; ) 1 0 A = 1 0 B = 0 1 2 1 Are the matrices similar? (iii) Consider the two 2 × 2 invertible matrices c - (; :). (: :) C = D = 1 Are the matrices similar?

Advanced Engineering Mathematics
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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3. Two n x n matrices A, B are called similar if there exists an invertible n x n matrix S such that A = SBS-1. If A is similar to B, then B is also similar to A, since B = S-'AS. (i) Show that the two 2 x 2 matrices A = (C ). 0 1 0 0 (: :) - B = 1 ĄT are similar. (ii) Consider the two 2 × 2 invertible matrices A- (; 9). B- (; 9) 1 0 B = 0 1 2 1 Are the matrices similar? (iii) Consider the two 2 × 2 invertible matrices 1 C = D- (: ) -1 1 Are the matrices similar?
Transcribed Image Text:3. Two n x n matrices A, B are called similar if there exists an invertible n x n matrix S such that A = SBS-1. If A is similar to B, then B is also similar to A, since B = S-'AS. (i) Show that the two 2 x 2 matrices A = (C ). 0 1 0 0 (: :) - B = 1 ĄT are similar. (ii) Consider the two 2 × 2 invertible matrices A- (; 9). B- (; 9) 1 0 B = 0 1 2 1 Are the matrices similar? (iii) Consider the two 2 × 2 invertible matrices 1 C = D- (: ) -1 1 Are the matrices similar?
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