第16章 16.19

(1)

$\begin{bmatrix} 1 & 2 & 3 & 4 & | & 2 \\ 2 & -1 & 1 & -1 & | & -4 \\ 3 & 1 & 4 & 3 & | & -2 \\ 1 & 3 & 4 & 5 & | & 2 \end{bmatrix} \to \begin{bmatrix} 1 & 2 & 3 & 4 & | & 2 \\ 0 & -5 & -5 & -9 & | & -8 \\ 0 & -5 & -5 & -9 & | & -8 \\ 0 & 1 & 1 & 1 & | & 0 \end{bmatrix} \to \begin{bmatrix} 1 & 2 & 3 & 4 & | & 2 \\ 0 & -5 & -5 & -9 & | & -8 \\ 0 & 0 & 0 & 0 & | & 0 \\ 0 & 1 & 1 & 1 & | & 0 \end{bmatrix}$

$\to \begin{bmatrix} 1 & 2 & 3 & 4 & | & 2 \\ 0 & 1 & 1 & 1 & | & 0 \\ 0 & 5 & 5 & 9 & | & 8 \\ 0 & 0 & 0 & 0 & | & 0 \end{bmatrix} \to \begin{bmatrix} 1 & 0 & 1 & 2 & | & 2 \\ 0 & 1 & 1 & 1 & | & 0 \\ 0 & 0 & 0 & 4 & | & 8 \\ 0 & 0 & 0 & 0 & | & 0 \end{bmatrix} \to \begin{bmatrix} 1 & 0 & 1 & 0 & | & -2 \\ 0 & 1 & 1 & 0 & | & -2 \\ 0 & 0 & 0 & 1 & | & 2 \\ 0 & 0 & 0 & 0 & | & 0 \end{bmatrix}$

よって $\begin{pmatrix} x \\ y \\ z \\ u \end{pmatrix} = k\begin{pmatrix} -1 \\ -1 \\ 1 \\ 0 \end{pmatrix} + \begin{pmatrix} -2 \\ -2 \\ 0 \\ 2 \end{pmatrix} \quad \text{（k は任意定数）}$

(2)

$\begin{bmatrix} 4 & 2 & 3 & 1 & | & 1 \\ 1 & 1 & 0 & -2 & | & 2 \\ 5 & 2 & 4 & -1 & | & 0 \end{bmatrix} \to \begin{bmatrix} 1 & 1 & 0 & -2 & | & 2 \\ 4 & 2 & 3 & 1 & | & 1 \\ 5 & 2 & 4 & -1 & | & 0 \end{bmatrix} \to \begin{bmatrix} 1 & 1 & 0 & -2 & | & 2 \\ 0 & -2 & 3 & 9 & | & -7 \\ 0 & -3 & 4 & 9 & | & -10 \end{bmatrix}$

$\to \begin{bmatrix} 1 & 0 & 1 & -2 & | & -1 \\ 0 & 1 & -1 & 0 & | & 3 \\ 0 & 0 & 1 & 9 & | & -1 \end{bmatrix} \to \begin{bmatrix} 1 & 0 & 0 & -11 & | & 0 \\ 0 & 1 & 0 & 9 & | & 2 \\ 0 & 0 & 1 & 9 & | & -1 \end{bmatrix}$

よって $\begin{pmatrix} x \\ y \\ z \\ w \end{pmatrix} = k\begin{pmatrix} 11 \\ -9 \\ -9 \\ 1 \end{pmatrix} + \begin{pmatrix} 0 \\ 2 \\ -1 \\ 0 \end{pmatrix} \quad \text{（k は任意定数）}$

(3)

$a = 1$のとき $x + y + z = 1$

$\left[\begin{array}{ccc|c} a & 1 & 1 & 1 \\ 1 & a & 1 & a \\ 1 & 1 & a & a^2 \end{array}\right] \to \left[\begin{array}{ccc|c} 1 & 1 & a & a^2 \\ 1 & a & 1 & a \\ a & 1 & 1 & 1 \end{array}\right] \to \left[\begin{array}{ccc|c} 1 & 1 & a & a^2 \\ 0 & a-1 & 1-a & a-a^2 \\ 0 & 1-a & 1-a^2 & 1-a^3 \end{array}\right] \to \left[\begin{array}{ccc|c} 1 & 1 & a & a^2 \\ 0 & 1 & -1 & -a \\ 0 & 1 & 1+a & 1+a+a^2 \end{array}\right]$

$\to \left[\begin{array}{ccc|c} 1 & 0 & a+1 & a^2+a \\ 0 & 1 & -1 & -a \\ 0 & 0 & a+2 & 1+2a+a^2 \end{array}\right] \to \left[\begin{array}{ccc|c} 1 & 0 & a+1 & a(a+1) \\ 0 & 1 & -1 & -a \\ 0 & 0 & 1 & (1+a)^2/(a+2) \end{array}\right]$

$\to \left[\begin{array}{ccc|c} 1 & 0 & 0 & a(a+1)-\frac{(1+a)^3}{a+2} \\ 0 & 1 & 0 & -a+\frac{(1+a)^2}{a+2} \\ 0 & 0 & 1 & \frac{(1+a)^2}{a+2} \end{array}\right] \to \left[\begin{array}{ccc|c} 1 & 0 & 0 & -\frac{a+1}{a+2} \\ 0 & 1 & 0 & \frac{1}{a+2} \\ 0 & 0 & 1 & \frac{(1+a)^2}{a+2} \end{array}\right]$

$a = 2$のとき解なし

$a \neq 1, a \neq 2$のとき

$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} -\frac{a+1}{a+2} \\ \frac{1}{a+2} \\ \frac{(1+a)^2}{a+2} \end{pmatrix}$