Find PA, MA for A = What about over Q? Justify your answers. Select one: 0 1 2 1 20 201 O over R. Is A diagonalisable? Over R we obtain P(x) = (x - √2)(x² + x - 3) which factors further to give 3 distinct real roots. Hence mA = PA and A is diagonalisable. Over Q there are no roots at all so mA = PA but A is not diagonalisable. O Over R we obtain P(x) = (x + 2)(x − √√3)² but (A +213) (A - √313) = 0 so m₁ = (x + 2)(x - √√3) and A is diagonalisable. But over Q we have PA(x) = (x − 3)(x² - 2x + 3) which does not factorise further and A 213 so mA = PA is not the product of distinct linear factors, so not diagonalisable. O Over R we obtain p₁(x) = (x − 3)(x − √3)(x + √3) which has 3 distinct roots. So mA = PA and A is diagonalisable. But over Q we just have PA(x) = (x − 3)(x² - 3) which does not factorise further and A 313 so mA = PA is not the product of distinct linear factors, so A is not diagonalisable. Over R we obtain P₁(x) = (x − 2)(x² + x + 3) which does not factor further, and A ‡ 213 so mд = PÅ is not the product of distinct linear factors, so A is not diagonalisable. The same holds over Q. None of the others apply
Find PA, MA for A = What about over Q? Justify your answers. Select one: 0 1 2 1 20 201 O over R. Is A diagonalisable? Over R we obtain P(x) = (x - √2)(x² + x - 3) which factors further to give 3 distinct real roots. Hence mA = PA and A is diagonalisable. Over Q there are no roots at all so mA = PA but A is not diagonalisable. O Over R we obtain P(x) = (x + 2)(x − √√3)² but (A +213) (A - √313) = 0 so m₁ = (x + 2)(x - √√3) and A is diagonalisable. But over Q we have PA(x) = (x − 3)(x² - 2x + 3) which does not factorise further and A 213 so mA = PA is not the product of distinct linear factors, so not diagonalisable. O Over R we obtain p₁(x) = (x − 3)(x − √3)(x + √3) which has 3 distinct roots. So mA = PA and A is diagonalisable. But over Q we just have PA(x) = (x − 3)(x² - 3) which does not factorise further and A 313 so mA = PA is not the product of distinct linear factors, so A is not diagonalisable. Over R we obtain P₁(x) = (x − 2)(x² + x + 3) which does not factor further, and A ‡ 213 so mд = PÅ is not the product of distinct linear factors, so A is not diagonalisable. The same holds over Q. None of the others apply
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.3: Quadratic Equations
Problem 81E
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