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.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.7: The Inverse Of A Matrix
Problem 32E
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