For a polynomial p(x) = anx" + an-1xn-1 +... + a₁x + ao with a; E F and square matrix A with entries in F, define p(A) by p(A) = an A+an-1A-1 + + a₁A+aoI. Let A € Mnxn(C) be diagonalizable and c(x) the characteristic polynomial of A. Then show (ie, prove) that c(A) = 0, where O is the n x n zero matrix.
For a polynomial p(x) = anx" + an-1xn-1 +... + a₁x + ao with a; E F and square matrix A with entries in F, define p(A) by p(A) = an A+an-1A-1 + + a₁A+aoI. Let A € Mnxn(C) be diagonalizable and c(x) the characteristic polynomial of A. Then show (ie, prove) that c(A) = 0, where O is the n x n zero matrix.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.3: Zeros Of Polynomials
Problem 4E
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![For a polynomial p(x) = anx" + an-1xn-¹ +... + a₁x + ao with a; E F and square matrix A
with entries in F, define p(A) by
n-1
p(A) = anA" + an-1 An + + a₁A + aoI.
Let A € Mnxn (C) be diagonalizable and c(x) the characteristic polynomial of A. Then show
(ie, prove) that c(A) = O, where O is the n x n zero matrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb04829d0-4645-426e-bf1a-7ada40b0786f%2F7b6c656b-4636-4f4e-ab03-ed26c99b4746%2F4pnozoo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For a polynomial p(x) = anx" + an-1xn-¹ +... + a₁x + ao with a; E F and square matrix A
with entries in F, define p(A) by
n-1
p(A) = anA" + an-1 An + + a₁A + aoI.
Let A € Mnxn (C) be diagonalizable and c(x) the characteristic polynomial of A. Then show
(ie, prove) that c(A) = O, where O is the n x n zero matrix.
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