Exercise 7.3.8

Answers

1.

Let

f (t)

be the characteristic polynomial of

T

. Recall that

\det (T) = f (0)

by Theorem 4.4.7. So By Theorem 7.12 and Theorem 7.14

0

is a zero for

p (t)

if and only if

0

is a zero for

f (t)

. Thus

T

is invertible if and only if

p (0) \neq 0

2.

Directly compute that

- \frac{1}{a_{0}} (T^{n - 1} + a_{n - 1} T^{n - 2} + \dots + a_{2} T + a_{1} I) T

= - \frac{1}{a_{0}} (p (T) - a_{0}) = I .

jephianlin

2011-06-27 00:00