高専の数学解説

第16章 16.6

(1)

(12232−2−373−414)→(12230−6−710−10−5−5)→(12230−6−710211)→(1−20108080211)\left(\begin{array}{ccc|c} 1 & 2 & 2 & 3 \\ 2 & -2 & -3 & 7 \\ 3 & -4 & 1 & 4 \end{array}\right) \to \left(\begin{array}{ccc|c} 1 & 2 & 2 & 3 \\ 0 & -6 & -7 & 1 \\ 0 & -10 & -5 & -5 \end{array}\right) \to \left(\begin{array}{ccc|c} 1 & 2 & 2 & 3 \\ 0 & -6 & -7 & 1 \\ 0 & 2 & 1 & 1 \end{array}\right) \to \left(\begin{array}{ccc|c} 1 & -2 & 0 & 1 \\ 0 & 8 & 0 & 8 \\ 0 & 2 & 1 & 1 \end{array}\right) (1−20101010211)→(10030101001−1)\left(\begin{array}{ccc|c} 1 & -2 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 2 & 1 & 1 \end{array}\right) \to \left(\begin{array}{ccc|c} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -1 \end{array}\right)

これから (x,y,z)=(3,1,−1)(x,y,z)=(3,1,-1)

(2)

(12343451657829)→(12340−2−440−3−79)→(1234012−20−3−79)→(10−18012−20−3−79)\left(\begin{array}{ccc|c} 1 & 2 & 3 & 4 \\ 3 & 4 & 5 & 16 \\ 5 & 7 & 8 & 29 \end{array}\right) \to \left(\begin{array}{ccc|c} 1 & 2 & 3 & 4 \\ 0 & -2 & -4 & 4 \\ 0 & -3 & -7 & 9 \end{array}\right) \to \left(\begin{array}{ccc|c} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & -2 \\ 0 & -3 & -7 & 9 \end{array}\right) \to \left(\begin{array}{ccc|c} 1 & 0 & -1 & 8 \\ 0 & 1 & 2 & -2 \\ 0 & -3 & -7 & 9 \end{array}\right) (10−18012−200−13)→(10050104001−3)\left(\begin{array}{ccc|c} 1 & 0 & -1 & 8 \\ 0 & 1 & 2 & -2 \\ 0 & 0 & -1 & 3 \end{array}\right) \to \left(\begin{array}{ccc|c} 1 & 0 & 0 & 5 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & -3 \end{array}\right)

これから (x,y,z)=(5,4,−3)(x,y,z)=(5,4,-3)

(3)

(5−12532−11374−531)→(5−12574−531−3−21−13)→(113031−8−60−34−3−21−13)→(226062−8−60−34−9−63−39)\left(\begin{array}{ccc|c} 5 & -1 & 2 & 5 \\ 3 & 2 & -1 & 13 \\ 7 & 4 & -5 & 31 \end{array}\right) \to \left(\begin{array}{ccc|c} 5 & -1 & 2 & 5 \\ 7 & 4 & -5 & 31 \\ -3 & -2 & 1 & -13 \end{array}\right) \to \left(\begin{array}{ccc|c} 11 & 3 & 0 & 31 \\ -8 & -6 & 0 & -34 \\ -3 & -2 & 1 & -13 \end{array}\right) \to \left(\begin{array}{ccc|c} 22 & 6 & 0 & 62 \\ -8 & -6 & 0 & -34 \\ -9 & -6 & 3 & -39 \end{array}\right) (140028−8−60−34−103−5)→(100243017−103−5)→(10020309003−3)→(10020103001−1)\left(\begin{array}{ccc|c} 14 & 0 & 0 & 28 \\ -8 & -6 & 0 & -34 \\ -1 & 0 & 3 & -5 \end{array}\right) \to \left(\begin{array}{ccc|c} 1 & 0 & 0 & 2 \\ 4 & 3 & 0 & 17 \\ -1 & 0 & 3 & -5 \end{array}\right) \to \left(\begin{array}{ccc|c} 1 & 0 & 0 & 2 \\ 0 & 3 & 0 & 9 \\ 0 & 0 & 3 & -3 \end{array}\right) \to \left(\begin{array}{ccc|c} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & -1 \end{array}\right)

これから

(x,y,z)=(2,3,−1)(x,y,z)=(2,3,-1)

(4)

(0523304201−204)→(1−204304200523)→(1−20406480523)→(1−2040−4020523)\left(\begin{array}{ccc|c} 0 & 5 & 2 & 3 \\ 3 & 0 & 4 & 20 \\ 1 & -2 & 0 & 4 \end{array}\right) \to \left(\begin{array}{ccc|c} 1 & -2 & 0 & 4 \\ 3 & 0 & 4 & 20 \\ 0 & 5 & 2 & 3 \end{array}\right) \to \left(\begin{array}{ccc|c} 1 & -2 & 0 & 4 \\ 0 & 6 & 4 & 8 \\ 0 & 5 & 2 & 3 \end{array}\right) \to \left(\begin{array}{ccc|c} 1 & -2 & 0 & 4 \\ 0 & -4 & 0 & 2 \\ 0 & 5 & 2 & 3 \end{array}\right)

(1−20∣4010∣−12052∣3)→(100∣3010∣−12002∣112)→(100∣3010∣−12001∣114)これから(x,y,z)=(3,−12,114)\begin{pmatrix} 1 & -2 & 0 & \Big| & 4 \\ 0 & 1 & 0 & \Big| & -\dfrac{1}{2} \\ 0 & 5 & 2 & \Big| & 3 \end{pmatrix} \rightarrow \begin{pmatrix} 1 & 0 & 0 & \Big| & 3 \\ 0 & 1 & 0 & \Big| & -\dfrac{1}{2} \\ 0 & 0 & 2 & \Big| & \dfrac{11}{2} \end{pmatrix} \rightarrow \begin{pmatrix} 1 & 0 & 0 & \Big| & 3 \\ 0 & 1 & 0 & \Big| & -\dfrac{1}{2} \\ 0 & 0 & 1 & \Big| & \dfrac{11}{4} \end{pmatrix} \quad \text{これから} \quad (x,y,z) = \left(3, -\dfrac{1}{2}, \dfrac{11}{4}\right)

解説: r31bn1z