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Linear Algebra

Exercise 5.4.18

Answers

1.

See Exercise 5.1.20.

2.

For simplicity, denote the right hand side to be

B

. Directly calculate that

AB = - \frac{1}{a_{0}} [{(- 1)}^{n} A^{n} + a_{n - 1} A^{n - 1} + \dots + a_{1} A]

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

3.

Calculate that the characteristic polynomial of

A

is

- t^{3} + 2 t^{2} + t - 2 .

So by the previous result, we know that

A^{- 1} = \frac{1}{2} [- A^{2} + 2 A + I]

= (\begin{matrix} 1 & - 1 & - 2 \\ 0 & \frac{1}{2} & \frac{3}{2} \\ 0 & 0 & - 1 \end{matrix}) .

jephianlin

2011-06-27 00:00

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