Exercise 1. Answer True or False. i) There exists a 3-by-3 matrix with real entries and no real eigenvalue. ii) There exists a non-diagonalizable 3-by-3 matrix whose characteristic polynomial is 23 -22- z. iii) If the two vectors u, v in R" are both perpendicular and linearly dependent then either u =0 or v= 0. iv) If A E M4,4(F) and rank(A) < 2 then A is similar to a matrix whose two most left columns are zero columns. v) There exists A E M3,5(F) which is diagonalizable and satisfies rank(A) = 1 and the sum of its eigenvalues is 0. %3D

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Exercise 1. Answer True or False.
i) There exists a 3-by-3 matrix with real entries and no real eigenvalue.
ii) There exists a non-diagonalizable 3-by-3 matrix whose characteristic polynomial is
23 - 22 - z.
iii) If the two vectors u, v in R" are both perpendicular and linearly dependent then
either u = 0 or v= 0.
iv) If A E M44(F) and rank(A) < 2 then A is similar to a matrix wh
columns are zero columns.
two most lef
v) There exists A E M3.5(F) which is diagonalizable and satisfies rank(A) = 1 and the
sum of its eigenvalues is 0.
Exercise 2. 1) If A E M3x3(F) has 1, 2 and 3 as eigenvalues, what is the dimension of
E(3, A)?
2) Consider
2 0
3.
Transcribed Image Text:1 / 2 125% Exercise 1. Answer True or False. i) There exists a 3-by-3 matrix with real entries and no real eigenvalue. ii) There exists a non-diagonalizable 3-by-3 matrix whose characteristic polynomial is 23 - 22 - z. iii) If the two vectors u, v in R" are both perpendicular and linearly dependent then either u = 0 or v= 0. iv) If A E M44(F) and rank(A) < 2 then A is similar to a matrix wh columns are zero columns. two most lef v) There exists A E M3.5(F) which is diagonalizable and satisfies rank(A) = 1 and the sum of its eigenvalues is 0. Exercise 2. 1) If A E M3x3(F) has 1, 2 and 3 as eigenvalues, what is the dimension of E(3, A)? 2) Consider 2 0 3.
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