Question

Question 5. Let A be a square matrix of order n and λ E R be an eigenvalue of A of geometric multiplicity k (1sks n) (a) Taking abasis Bo of EAA the eigenspace of A for the eigenvalue λ and extending it to a basis B of R" show that MatB+a(A)-(Ολ4.P), for 80m e matrices P of order k × (n-k) and Q of order (n-k) x (n-k), where 4 is the identity matrix of order k and On-사 ia the zero matrix of order (n-k) × k. [10 marks (b) Show that the characteristic polynomial of Matg-.B(A) 1S: where q(z) E R] is the characteristic polynomial of the square matrıx (10 marks)

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Answer 1

Given B = {1, 1 + x, 1 + x, + x^2} Let a + bx + cx^2 elementof P_2. We will show that a + bx + cz^2 = alpha middot 1 + beta (1 + x) + gamma (1 + x + x^2) 1 for scalars alpha, beta gamma and also B is linearly independent a + bx + cx^2 = alpha middot 1 + beat (1 + x) + r (1 + x + x^2) = > a + bx + cx^2 = (alpha + beat + gamma) + (beta + gamma) x + gamma x^2 comparing coefficient of like terms both sides gamma = c, beta + gamma = b, alpha + beat + gamma = a beta = b - gamma = b - c, alpha = a - beta - gamma = a - (b - c) - c Thus alpha = a - b, beta = b - c, gamma = c = > B spans p_2