1. Let A = and B = For what values of t is 2A+B invertible? 2. Let n be a natural number and let A, B E M,xn(C). If AB = 0, does it follow that either A 0 or B = 0? Justify your answer. 3. Suppose that BE Mnxn(C) and B3 = 0. Show that I-B is invertible. Hint: Consider I+ B+ B2. 4. Let GL2(C) denote the set of invertible matrices in M2x2(C). Does GL2(C) form a complex vector space under the usual addition and multiplication defined on 2 by 2 matrices?
1. Let A = and B = For what values of t is 2A+B invertible? 2. Let n be a natural number and let A, B E M,xn(C). If AB = 0, does it follow that either A 0 or B = 0? Justify your answer. 3. Suppose that BE Mnxn(C) and B3 = 0. Show that I-B is invertible. Hint: Consider I+ B+ B2. 4. Let GL2(C) denote the set of invertible matrices in M2x2(C). Does GL2(C) form a complex vector space under the usual addition and multiplication defined on 2 by 2 matrices?
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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Transcribed Image Text:1. Let
A =
3
and
B =
For what values of t is 2A +B invertible?
2. Let n be a natural number and let A, BE Mnxn(C). If AB = 0, does it follow that either A =0 or
B= 0? Justify your answer.
3. Suppose that BE Mnxn(C) and B3 = 0. Show that I-B is invertible. Hint: Consider I+ B+ B2.
4. Let GL2(C) denote the set of invertible matrices in M2x2(C). Does GL2(C) form a complex vector
space under the usual addition and multiplication defined on 2 by 2 matrices?
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