Let a and b be real numbers satisfying a² - a -1 = 0 and 6² - 36 - 2 = 0, and let V = 1 a b ab (a) Find a 4×4 matrix A with integer entries such that Av = av, and find a 4 × 4 matrix B with integer entries such that Bv = bv. Hint: Note that a² = 1+a and b² = 2+3b. Do not try to use the quadratic formula to express a and b in terms of square roots, because that formula will not help here. (b) Show that (A+B)v = (a+b)v and that ABv = bav = abv. Take care with the order of multiplication when treating ABv. (c) Deduce from part (b) that there are polynomials f(x) = x² + 3x³ + ₂x² + ₁x + co g(x) = x² + d3x³ + d₂x² + d₁x + do such that f(a + b) = 0 and g(ab) = 0, where Co,..., C3, do, ..., dz are integers. You do not need to find f(x) and g(x) explicitly. Hint: Consider characteristic polynomials.
Let a and b be real numbers satisfying a² - a -1 = 0 and 6² - 36 - 2 = 0, and let V = 1 a b ab (a) Find a 4×4 matrix A with integer entries such that Av = av, and find a 4 × 4 matrix B with integer entries such that Bv = bv. Hint: Note that a² = 1+a and b² = 2+3b. Do not try to use the quadratic formula to express a and b in terms of square roots, because that formula will not help here. (b) Show that (A+B)v = (a+b)v and that ABv = bav = abv. Take care with the order of multiplication when treating ABv. (c) Deduce from part (b) that there are polynomials f(x) = x² + 3x³ + ₂x² + ₁x + co g(x) = x² + d3x³ + d₂x² + d₁x + do such that f(a + b) = 0 and g(ab) = 0, where Co,..., C3, do, ..., dz are integers. You do not need to find f(x) and g(x) explicitly. Hint: Consider characteristic polynomials.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.3: Matrix Algebra
Problem 89E
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Transcribed Image Text:Let a and b be real numbers satisfying a² - a -1 = 0 and 6² - 36 - 2 = 0,
and let
V =
1
a
b
ab
(a) Find a 4×4 matrix A with integer entries such that Av = av, and find
a 4 × 4 matrix B with integer entries such that Bv = bv. Hint: Note
that a² = 1+a and b² = 2+3b. Do not try to use the quadratic formula
to express a and b in terms of square roots, because that formula will
not help here.
(b) Show that (A+B)v = (a+b)v and that ABv = bav = abv. Take care
with the order of multiplication when treating ABv.
(c) Deduce from part (b) that there are polynomials
f(x) = x² + 3x³ + ₂x² + ₁x + co
g(x) = x² + d3x³ + d₂x² + d₁x + do
such that f(a + b) = 0 and g(ab) = 0, where Co,..., C3, do, ..., dz
are integers. You do not need to find f(x) and g(x) explicitly. Hint:
Consider characteristic polynomials.
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