高専の数学解説

第16章 16.18

(1)

{x−y+2z=−1−x−y−2z=34x+3y+3z=−1\begin{cases} x - y + 2z = -1 \\ -x - y - 2z = 3 \\ 4x + 3y + 3z = -1 \end{cases} [1−12−1−1−1−23433−1]→[1−12−10−20207−53]→[1−12−1010−107−53]→[102−2010−100−510]→[102−2010−1001−2]\left[\begin{array}{ccc|c} 1 & -1 & 2 & -1 \\ -1 & -1 & -2 & 3 \\ 4 & 3 & 3 & -1 \end{array}\right] \rightarrow \left[\begin{array}{ccc|c} 1 & -1 & 2 & -1 \\ 0 & -2 & 0 & 2 \\ 0 & 7 & -5 & 3 \end{array}\right] \rightarrow \left[\begin{array}{ccc|c} 1 & -1 & 2 & -1 \\ 0 & 1 & 0 & -1 \\ 0 & 7 & -5 & 3 \end{array}\right] \rightarrow \left[\begin{array}{ccc|c} 1 & 0 & 2 & -2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & -5 & 10 \end{array}\right] \rightarrow \left[\begin{array}{ccc|c} 1 & 0 & 2 & -2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -2 \end{array}\right] →[1002010−1001−2]\rightarrow \left[\begin{array}{ccc|c} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -2 \end{array}\right]

よって

(xyz)=(2−1−2)\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \\ -2 \end{pmatrix}

(2)

{2x−y+z=34x+2y+3z=5x+y+z=2\begin{cases} 2x - y + z = 3 \\ 4x + 2y + 3z = 5 \\ x + y + z = 2 \end{cases} [2−11342351112]→[111242352−113]→[11120−2−1−30−3−1−1]→[1−2010−2−1−30311]\left[\begin{array}{ccc|c} 2 & -1 & 1 & 3 \\ 4 & 2 & 3 & 5 \\ 1 & 1 & 1 & 2 \end{array}\right] \rightarrow \left[\begin{array}{ccc|c} 1 & 1 & 1 & 2 \\ 4 & 2 & 3 & 5 \\ 2 & -1 & 1 & 3 \end{array}\right] \rightarrow \left[\begin{array}{ccc|c} 1 & 1 & 1 & 2 \\ 0 & -2 & -1 & -3 \\ 0 & -3 & -1 & -1 \end{array}\right] \rightarrow \left[\begin{array}{ccc|c} 1 & -2 & 0 & 1 \\ 0 & -2 & -1 & -3 \\ 0 & 3 & 1 & 1 \end{array}\right] →[100−3010−20017]\rightarrow \left[\begin{array}{ccc|c} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 7 \end{array}\right]

よって

(xyz)=(−3−27)\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} -3 \\ -2 \\ 7 \end{pmatrix}

解説: r31bn1z