10/10 作业

Homework 8

1. $m \ne -2$ or $1$ , sol. = $\begin{pmatrix}\mathop{-}\left( \frac{1}{m\mathop{+}2}\right) \\ \frac{m\mathop{+}1}{m\mathop{+}2}\\ \frac{m\mathop{+}1}{m\mathop{+}2}\end{pmatrix}$ .

2. $A^* = \begin{pmatrix}3 & 2 & 1\\ 2 & 4 & 2\\ 1 & 2 & 3\end{pmatrix}$ , $A^{-1} = \begin{pmatrix}\frac{3}{4} & \frac{1}{2} & \frac{1}{4}\\ \frac{1}{2} & 1 & \frac{1}{2}\\ \frac{1}{4} & \frac{1}{2} & \frac{3}{4}\end{pmatrix}$ .

3. $A^* = \begin{pmatrix}0 & 0 & 0 & \mathop{-}1\\ 0 & \mathop{-}1 & 0 & 0\\ 0 & 0 & \mathop{-}1 & 0\\ \mathop{-}1 & 0 & 0 & 0\end{pmatrix}$ , $B^* = \begin{pmatrix}24 & 0 & 0 & 0\\ 0 & 12 & 0 & 0\\ 0 & 0 & 8 & 0\\ 0 & 0 & 0 & 6\end{pmatrix}$ .

4. (1) $|A|^2 A$ .

   (2) Inverse.

   (3) $(AB)^* = BA$ .

Optional

1. $(dx_1, dy_1, dp) = \begin{pmatrix} - \frac{\left( {R_1}\mathop{-}1\right) {R_2} \left( \alpha\mathop{-}1\right) \alpha d {e_2}}{{R_2} \alpha\mathop{-}{R_1} \alpha\mathop{+}{R_1}}\\ -\frac{{R_2} \alpha \left( {R_1} \alpha\mathop{-}\alpha\mathop{-}{R_1}\right) d {e_2}}{{R_2} \alpha\mathop{-}{R_1} \alpha\mathop{+}{R_1}}\\ -\frac{{R_1} {R_2} \alpha d {e_2}}{\left( \alpha\mathop{-}1\right) \, \left( {R_2} \alpha\mathop{-}{R_1} \alpha\mathop{+}{R_1}\right) }\end{pmatrix}$ .

   If $de_2 > 0$ , then $dy_1 > 0$ and $dp > 0$ .