第16章 16.22

(1)

$\begin{vmatrix} a & b & 0 \\ c & d & 0 \\ 0 & 0 & c \end{vmatrix} = acd - bc^2 = c(ad - bc) \neq 0$

(2)

$\begin{vmatrix} d & 0 \\ 0 & c \end{vmatrix} = cd \quad \begin{vmatrix} c & 0 \\ 0 & c \end{vmatrix} = c^2 \quad \begin{vmatrix} c & d \\ 0 & 0 \end{vmatrix} = 0$

$\begin{vmatrix} b & 0 \\ 0 & c \end{vmatrix} = bc \quad \begin{vmatrix} a & 0 \\ 0 & c \end{vmatrix} = ac \quad \begin{vmatrix} a & b \\ 0 & 0 \end{vmatrix} = 0$

$\begin{vmatrix} b & 0 \\ d & 0 \end{vmatrix} = 0 \quad \begin{vmatrix} a & 0 \\ c & 0 \end{vmatrix} = 0 \quad \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$

これから

$A^{-1} = \frac{1}{c(ad-bc)} \begin{pmatrix} cd & -bc & 0 \\ -c^2 & ac & 0 \\ 0 & 0 & ad - bc \end{pmatrix} = \frac{1}{ad - bc} \begin{pmatrix} d & -b & 0 \\ -c & a & 0 \\ 0 & 0 & \frac{ad-bc}{c} \end{pmatrix}$