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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please answer 21
![**Exercise 20:**
Let \( A \) be an \( n \times n \) matrix with characteristic polynomial
\[
f(t) = (-1)^n t^n + a_{n-1} t^{n-1} + \cdots + a_1 t + a_0.
\]
- Prove that \( f(0) = a_0 = \det(A) \). Deduce that \( A \) is invertible if and only if \( a_0 \neq 0 \).
**Exercise 21:**
Let \( A \) and \( f(t) \) be as in Exercise 20.
(a) Prove that
\[
f(t) = (A_{11} - t)(A_{22} - t) \cdots (A_{nn} - t) + q(t),
\]
where \( q(t) \) is a polynomial of degree at most \( n-2 \). *Hint: Apply mathematical induction to \( n \).*
(b) Show that
\[
\text{tr}(A) = (-1)^{n-1} a_{n-1}.
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Transcribed Image Text:**Exercise 20:**
Let \( A \) be an \( n \times n \) matrix with characteristic polynomial
\[
f(t) = (-1)^n t^n + a_{n-1} t^{n-1} + \cdots + a_1 t + a_0.
\]
- Prove that \( f(0) = a_0 = \det(A) \). Deduce that \( A \) is invertible if and only if \( a_0 \neq 0 \).
**Exercise 21:**
Let \( A \) and \( f(t) \) be as in Exercise 20.
(a) Prove that
\[
f(t) = (A_{11} - t)(A_{22} - t) \cdots (A_{nn} - t) + q(t),
\]
where \( q(t) \) is a polynomial of degree at most \( n-2 \). *Hint: Apply mathematical induction to \( n \).*
(b) Show that
\[
\text{tr}(A) = (-1)^{n-1} a_{n-1}.
\]
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