(a) Let p(x) be a polynomial with coefficients in R, and let M abe the 3 x 3 matrix а 1 Ma := a 1 а Show that p(a) p'(a) p"(a) p(a) p'(a) p(a). p(Ma) (Hint: prove it first for monomials of the form x" by induction on n, and use the principle of linearity to prove this for all polynomials.) (b) Use this to determine the minimal polynomial of Ma.

(a) Let p(x) be a polynomial with coefficients in R, and let M abe the 3 x 3 matrix а 1 Ma := a 1 а Show that p(a) p'(a) p"(a) p(a) p'(a) p(a). p(Ma) (Hint: prove it first for monomials of the form x" by induction on n, and use the principle of linearity to prove this for all polynomials.) (b) Use this to determine the minimal polynomial of Ma.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.1: Polynomials Over A Ring

Problem 12E: a. Find a nonconstant polynomial in Z4[ x ], if one exists, that is a unit. b. Find a nonconstant...

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