12/25 作业

Homework 17

1. (a) None.

   (b) $\lambda = \pm i$ .

2. Yes.

3. (1) $D = \begin{pmatrix} 2 & 0 \\ 0 & 8 \end{pmatrix}$ , $P = \begin{pmatrix} -1 & 1 \\ 1 & 1 \end{pmatrix}$ .

   (2) $D = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}$ , $P = \begin{pmatrix} 1 & 2 & 1 \\ 1 & 3 & 3 \\ 1 & 3 & 4 \end{pmatrix}$ .

   (3) Not diagonalizable.

4. (1) No.

   (2) Yes.

   (3) Not determined.

5. $A^k = \begin{pmatrix} a^k & 0 \\ 3a^k - 3b^k & b^k \end{pmatrix}$ .

6. (1) True.

   (2) True.

   (3) True.

   (4) True.

   (5) False.

   (6) True.