13. a) Show that if A and B are n x n matrices, at least one of which is invertible, then AB and BA are similar. b) What do the matrices A = and B = %3D have to do with this issue?
13. a) Show that if A and B are n x n matrices, at least one of which is invertible, then AB and BA are similar. b) What do the matrices A = and B = %3D have to do with this issue?
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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![13. a) Show that if A and B arenxn matrices, at least one of which is
invertible, then AB and BA are similar.
b) What do the matrices
and B- : ]
A
have to do with this issue?
c) Show that even without the assumption on invertibility the matrices
AB and BA have the same characteristic polynomial. Hint: Write
A = PIQ
where P and Q are invertible and I, is an n x n matrix that has the
rxr identity in the upper left-hand corner and 0's elsewhere. Write
B' = QBP. Compute AB and BA and find their characteristic
polynomials.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb4af5bf-9b75-4cca-a55b-ef482e95bc73%2F9b950227-3a32-4a3c-bb97-53c6f4a47007%2Fa9a7f9n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:13. a) Show that if A and B arenxn matrices, at least one of which is
invertible, then AB and BA are similar.
b) What do the matrices
and B- : ]
A
have to do with this issue?
c) Show that even without the assumption on invertibility the matrices
AB and BA have the same characteristic polynomial. Hint: Write
A = PIQ
where P and Q are invertible and I, is an n x n matrix that has the
rxr identity in the upper left-hand corner and 0's elsewhere. Write
B' = QBP. Compute AB and BA and find their characteristic
polynomials.
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