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(1) Let T: R3 → M2x2 be a linear map that is injective. Then T is also surjective.

(1) Let T: R3 → M2x2 be a linear map that is injective. Then T is also surjective.

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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Problem 7 Prove or disprove the following statements: (1) Let T: R3 → M2x2 be a linear map that is injective. Then T is also surjective. (2) A is an n x n matrix, such that 2A3 – 4A? + 6A – 21 = 0 (where I is the identity matrix and O is the zero matrix) then A is non singular. (3) If A and B are 3 x 3 matrices such that det(A) = 5 and det(B) = 4, Then det(A" (5B)-1) = 100 3 0 0 1 4 0 5 0 4 40 0 0 3 0 0 4 (4) Consider the following 2 matrices: A = and B = Then A is similar to B.
Transcribed Image Text:Problem 7 Prove or disprove the following statements: (1) Let T: R3 → M2x2 be a linear map that is injective. Then T is also surjective. (2) A is an n x n matrix, such that 2A3 – 4A? + 6A – 21 = 0 (where I is the identity matrix and O is the zero matrix) then A is non singular. (3) If A and B are 3 x 3 matrices such that det(A) = 5 and det(B) = 4, Then det(A" (5B)-1) = 100 3 0 0 1 4 0 5 0 4 40 0 0 3 0 0 4 (4) Consider the following 2 matrices: A = and B = Then A is similar to B.
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