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Math 369 HW #8 Due at the beginning of class on Friday, October 31 Reading: Sections DM, SD. Problems: 1. Let 2 2 0 A = 2 2 0 . 0 0 4 (a) Find numbers a0 , a1 , a2 (not all zero) so that a0 13 + a1 A + a2 A2 = 0. (b) Determine the minimal polynomial minpolyA (x) of A (don’t forget to show that it has minimal degree!). 2. Suppose B is an n × n matrix and let s ∈ R be any number. Prove that * * * { x ∈ Rn : B x = s x} is a subspace of Rn . 3. Let A be the same matrix as in problem #1. (a) Final all numbers s so that minpolyA (s) = 0. (b) For each s from part (a), determine * * * Vs = { v ∈ R3 : A v = s v }. Find a basis for Vs (by Problem 2, you know it’s a subspace of R3 , so it must have a basis). 4. Repeat problems 1 and 3 with the matrix 1 −12 e 1 0 . B = 0 0 0 1 1