高専の数学解説

第18章 18.15

(1)

[12−13−212413−32−1−22−4−113606−53]→[12−13−21003−310001−1−32003−310]→[12−13−21001−1−32003−310003−310]\left[\begin{array}{ccccc|c} 1 & 2 & -1 & 3 & -2 & 1 \\ 2 & 4 & 1 & 3 & -3 & 2 \\ -1 & -2 & 2 & -4 & -1 & 1 \\ 3 & 6 & 0 & 6 & -5 & 3 \end{array}\right] \to \left[\begin{array}{ccccc|c} 1 & 2 & -1 & 3 & -2 & 1 \\ 0 & 0 & 3 & -3 & 1 & 0 \\ 0 & 0 & 1 & -1 & -3 & 2 \\ 0 & 0 & 3 & -3 & 1 & 0 \end{array}\right] \to \left[\begin{array}{ccccc|c} 1 & 2 & -1 & 3 & -2 & 1 \\ 0 & 0 & 1 & -1 & -3 & 2 \\ 0 & 0 & 3 & -3 & 1 & 0 \\ 0 & 0 & 3 & -3 & 1 & 0 \end{array}\right]

→[1202−53001−1−32000010−6000000]→[1202−53001−10−2300001−35000000]→[120200001−10200001−35000000]\to \left[\begin{array}{ccccc|c} 1 & 2 & 0 & 2 & -5 & 3 \\ 0 & 0 & 1 & -1 & -3 & 2 \\ 0 & 0 & 0 & 0 & 10 & -6 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \to \left[\begin{array}{ccccc|c} 1 & 2 & 0 & 2 & -5 & 3 \\ 0 & 0 & 1 & -1 & 0 & -\frac{2}{3} \\ 0 & 0 & 0 & 0 & 1 & -\frac{3}{5} \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right] \to \left[\begin{array}{ccccc|c} 1 & 2 & 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 2 \\ 0 & 0 & 0 & 0 & 1 & -\frac{3}{5} \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right]

これから

[x1x2x3x4x5]=C1[−21000]+C2[−20110]+[0020−35]\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \end{bmatrix} = C_1 \begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} + C_2 \begin{bmatrix} -2 \\ 0 \\ 1 \\ 1 \\ 0 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 2 \\ 0 \\ -\frac{3}{5} \end{bmatrix}

（C1,C2C_1, C_2は任意）

(2)

[11234523910743217111341−751518]→[1123450154−1−60−1−5−2−1−20−3−15−7−1−2]→[10−3−15110154−1−60002−2−80005−4−20]\left[\begin{array}{ccccc|c} 1 & 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 9 & 10 & 7 & 4 \\ 3 & 2 & 1 & 7 & 11 & 13 \\ 4 & 1 & -7 & 5 & 15 & 18 \end{array}\right] \to \left[\begin{array}{ccccc|c} 1 & 1 & 2 & 3 & 4 & 5 \\ 0 & 1 & 5 & 4 & -1 & -6 \\ 0 & -1 & -5 & -2 & -1 & -2 \\ 0 & -3 & -15 & -7 & -1 & -2 \end{array}\right] \to \left[\begin{array}{ccccc|c} 1 & 0 & -3 & -1 & 5 & 11 \\ 0 & 1 & 5 & 4 & -1 & -6 \\ 0 & 0 & 0 & 2 & -2 & -8 \\ 0 & 0 & 0 & 5 & -4 & -20 \end{array}\right]

→[10−3−15110154−1−60001−1−40005−4−20]→[10−304701503100001−1−4000010]→[10−3007015001000010−4000010]\to \left[\begin{array}{ccccc|c} 1 & 0 & -3 & -1 & 5 & 11 \\ 0 & 1 & 5 & 4 & -1 & -6 \\ 0 & 0 & 0 & 1 & -1 & -4 \\ 0 & 0 & 0 & 5 & -4 & -20 \end{array}\right] \to \left[\begin{array}{ccccc|c} 1 & 0 & -3 & 0 & 4 & 7 \\ 0 & 1 & 5 & 0 & 3 & 10 \\ 0 & 0 & 0 & 1 & -1 & -4 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right] \to \left[\begin{array}{ccccc|c} 1 & 0 & -3 & 0 & 0 & 7 \\ 0 & 1 & 5 & 0 & 0 & 10 \\ 0 & 0 & 0 & 1 & 0 & -4 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right]

これから

[x1x2x3x4x5]=C1[3−5100]+[7100−40]\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \end{bmatrix} = C_1 \begin{bmatrix} 3 \\ -5 \\ 1 \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} 7 \\ 10 \\ 0 \\ -4 \\ 0 \end{bmatrix}

解説: r31bn1z