0 i 0 Let A = 1 10 as a matrix over C, where i = √√-1. Li 0 i Find p₁(x), m₁(x) and determine if A is diagonalisable. Justify your answer. Select one: O None of the others apply O PA(x) = (x + i)(x − i)² but (A + iI3) (A - iI3) = 0 so m₁(x) = (x + i)(x - i) which is a product of distinct linear factors, so A is diagonalisable. O PA₁(x) = (x + i)(x² + √2x+i) which does not factorise further. Since A -13 it follows that mA = PA which is not the product of distinct linear factors, so A is not diagonalisable. ○ P₁(x) = (xi)(x² - x - i) which factorises further as it must over C. These 3 roots are all distinct so m₁ = PA is a product of distinct linear factors, so A is diagonalisable. O P₁(x) = (xi)(x + i)² but (A − iI3) (A + iI3) #0 so m₁ = PA which is not a product of distinct linear factors, so A is not diagonalisable.
0 i 0 Let A = 1 10 as a matrix over C, where i = √√-1. Li 0 i Find p₁(x), m₁(x) and determine if A is diagonalisable. Justify your answer. Select one: O None of the others apply O PA(x) = (x + i)(x − i)² but (A + iI3) (A - iI3) = 0 so m₁(x) = (x + i)(x - i) which is a product of distinct linear factors, so A is diagonalisable. O PA₁(x) = (x + i)(x² + √2x+i) which does not factorise further. Since A -13 it follows that mA = PA which is not the product of distinct linear factors, so A is not diagonalisable. ○ P₁(x) = (xi)(x² - x - i) which factorises further as it must over C. These 3 roots are all distinct so m₁ = PA is a product of distinct linear factors, so A is diagonalisable. O P₁(x) = (xi)(x + i)² but (A − iI3) (A + iI3) #0 so m₁ = PA which is not a product of distinct linear factors, so A is not diagonalisable.
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:0 i 0
1 1 0 as a matrix over C, where i = √-1.
i
0 i
Find p₁(x), m₁ (x) and determine if A is diagonalisable. Justify your answer.
Let A =
Select one:
None of the others apply
PA(x) = (x + i)(x − i)² but (A + iI3)(A − iI3) = 0 som₁(x) = (x + i)(x - i) which is a product of distinct linear factors, so A is diagonalisable.
P₁(x) = (x + i)(x² + √2x + i) which does not factorise further. Since A -I3 it follows that m₁ = PA which is not the product of distinct linear
factors, so A is not diagonalisable.
○ P₁(x) = (xi)(x² − x − i) which factorises further as it must over C. These 3 roots are all distinct so m₁ = P₁ is a product of distinct linear factors, so A
is diagonalisable.
O
P₁(x) = (x − i)(x + i)² but (A − iI3) (A + iI3) ‡ 0 so m₁ = PA which is not a product of distinct linear factors, so A is not diagonalisable.
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