Let A be an n x n matrix with characteristic polynomial f(t) = (-1)"t" + an-1t²-1 +...+ a₁t + ao. Prove that f(0) = ao = det(A). Deduce that A is invertib if ao # 0.
Let A be an n x n matrix with characteristic polynomial f(t) = (-1)"t" + an-1t²-1 +...+ a₁t + ao. Prove that f(0) = ao = det(A). Deduce that A is invertib if ao # 0.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices
Problem 28EQ
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Question
Please answer 21
Transcribed Image Text:20. Let A be an n x n matrix with characteristic polynomial
f(t)= (-1)""+an-it"-1 ++at+ao.
Prove that f(0) = ao = det(A). Deduce that A is invertible if and only
if ao # 0.
21. Let A and f(t) be as in Exercise 20.
(a)
Prove that f(t) = (A11-t) (A22-t) (Ann-t)+q(t), where q(t)
is a polynomial of degree at most n-2. Hint: Apply mathematical
induction to n.
(b) Show that tr(A) = (-1)-¹an-1.
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