Question

Problem 2. For each polynomial p(t) = do +at+...+ amtm with real number coefficients and for each n x n matrix A, we define the n x n matrix p(A) by P(A) = ao In + a A+ ... + amA”. Also, for each n, let Onxn E Rnxn be the n x n zero matrix. (a) Show that for all polynomials p and q and square matrices A, we have p(A)q(A) = 9(A)p(A). (b) Show that for every 2 x 2 matrix A, we have A2 – (trA)A + (detA)12 = 02x2. (c) Let A be a 2 x 2 matrix and let n E N. Show that the subspace span(12, A, ..., An) of R2x2 has dimension at most two. (d) Show that for every nxn matrix A, there exists a non-zero polynomial p such that p(A) = Onxn (e) Let p(t) be any polynomial and let A be an n x n matrix such that p(A) = 0nxn. Show that p(1) = 0 for every eigenvalue 1 of A.

Answer 1

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