作业 0908

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  1. (1) Aug. = (24−2249−38−2−3710)\begin{pmatrix}2 & 4 & \mathop{-}2 & 2\\ 4 & 9 & \mathop{-}3 & 8\\ \mathop{-}2 & \mathop{-}3 & 7 & 10\end{pmatrix} .

    REF = (12−1101140012)\begin{pmatrix}1 & 2 & \mathop{-}1 & 1\\ 0 & 1 & 1 & 4\\ 0 & 0 & 1 & 2\end{pmatrix} .

    RREF = (100−101020012)\begin{pmatrix}1 & 0 & 0 & \mathop{-}1\\ 0 & 1 & 0 & 2\\ 0 & 0 & 1 & 2\end{pmatrix} .

    (x1,x2,x3)=(−1,2,2)(x_1,x_2,x_3)=(-1,2,2) .

    (2) Aug. = (1−34−43−77−8−46−17)\begin{pmatrix}1 & \mathop{-}3 & 4 & \mathop{-}4\\ 3 & \mathop{-}7 & 7 & \mathop{-}8\\ \mathop{-}4 & 6 & \mathop{-}1 & 7\end{pmatrix} .

    REF = (1−34−401−5220001)\begin{pmatrix}1 & \mathop{-}3 & 4 & \mathop{-}4\\ 0 & 1 & -\frac{5}{2} & 2\\ 0 & 0 & 0 & 1\end{pmatrix} .

    无解。

    (3) Aug. = (34−1868−23)\begin{pmatrix}3 & 4 & \mathop{-}1 & 8\\ 6 & 8 & \mathop{-}2 & 3\end{pmatrix} .

    REF = (143−13830001)\begin{pmatrix}1 & \frac{4}{3} & -\frac{1}{3} & \frac{8}{3}\\ 0 & 0 & 0 & 1\end{pmatrix} .

    无解。

    (4) Aug. = (1112−15342)\begin{pmatrix}1 & 1 & 1\\ 2 & \mathop{-}1 & 5\\ 3 & 4 & 2\end{pmatrix} .

    REF = (11101−1000)\begin{pmatrix}1 & 1 & 1\\ 0 & 1 & \mathop{-}1\\ 0 & 0 & 0\end{pmatrix} .

    RREF = (10201−1000)\begin{pmatrix}1 & 0 & 2\\ 0 & 1 & \mathop{-}1\\ 0 & 0 & 0\end{pmatrix} .

    (x,y)=(2,−1)(x,y)=(2,-1) .

    (5) Aug. = (40−3−693−1−2−113−2−30−1−9145−7)\begin{pmatrix}4 & 0 & \mathop{-}3 & \mathop{-}6 & 9\\ 3 & \mathop{-}1 & \mathop{-}2 & \mathop{-}1 & 1\\ 3 & \mathop{-}2 & \mathop{-}3 & 0 & \mathop{-}1\\ \mathop{-}9 & 1 & 4 & 5 & \mathop{-}7\end{pmatrix} .

    REF = (10−34−329401−14−722340012−300000)\begin{pmatrix}1 & 0 & -\frac{3}{4} & -\frac{3}{2} & \frac{9}{4}\\ 0 & 1 & -\frac{1}{4} & -\frac{7}{2} & \frac{23}{4}\\ 0 & 0 & 1 & 2 & -3\\ 0 & 0 & 0 & 0 & 0\end{pmatrix} .

    {x=5+3zy=−3−2zw=0\begin{cases}x = 5+3z \\ y=-3-2z \\ w = 0 \end{cases} , zz 自由变量。

    (6) Aug. = (10010001100110001010)\begin{pmatrix}1 & 0 & 0 & 1 & 0\\ 0 & 0 & 1 & 1 & 0\\ 0 & 1 & 1 & 0 & 0\\ 0 & 1 & 0 & 1 & 0\end{pmatrix} .

    REF = (10010011000011000010)\begin{pmatrix}1 & 0 & 0 & 1 & 0\\ 0 & 1 & 1 & 0 & 0\\ 0 & 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 1 & 0\end{pmatrix} .

    RREF = (10000010000010000010)\begin{pmatrix}1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\end{pmatrix} .

    (w,x,y,z)=(0,0,0,0)(w,x,y,z)=(0,0,0,0) .

  2. (1) (1−47g03−5h0−35k−2g)\begin{pmatrix}1 & \mathop{-}4 & 7 & g\\ 0 & 3 & \mathop{-}5 & h\\ 0 & \mathop{-}3 & 5 & k\mathop{-}2 g\end{pmatrix} , h=k−2gh=k-2g .

    (2) M3,∗=M1,∗−M2,∗M_{3,*}=M_{1,*}-M_{2,*} , a+1=2−(−1)a+1=2-(-1) , a=2a=2 .

  3. 44 , 44 , 66 .

  4. 都存在。行操作都有逆元。

  5. Aug. = (111p124q139r)\begin{pmatrix}1 & 1 & 1 & p\\ 1 & 2 & 4 & q\\ 1 & 3 & 9 & r\end{pmatrix} .

    REF = (111p013q−p001r−2q+p2)\begin{pmatrix}1 & 1 & 1 & p\\ 0 & 1 & 3 & q\mathop{-}p\\ 0 & 0 & 1 & \frac{r\mathop{-}2 q\mathop{+}p}{2}\end{pmatrix} .

    有解。

  6. Aug. = (0.20.5210001111000)\begin{pmatrix}0.2 & 0.5 & 2 & 1000\\ 1 & 1 & 1 & 1000\end{pmatrix} .

    REF = (15210500001680003)\begin{pmatrix}1 & \frac{5}{2} & 10 & 5000\\ 0 & 1 & 6 & \frac{8000}{3}\end{pmatrix}

    RREF = (10−5−5000301680003)\begin{pmatrix}1 & 0 & -5 & -\frac{5000}{3} \\ 0 & 1 & 6 & \frac{8000}{3}\end{pmatrix} .

    3y+18z=80003y + 18z = 8000 无整数解。

    无解。

  7. {x1+x3=20x1+x2=80x3+x4−x2=0\begin{cases} x_1 + x_3 = 20 \\ x_1 + x_2 = 80 \\ x_3 + x_4 - x_2 = 0 \end{cases} .

    Aug. = (1010201100800−1110)\begin{pmatrix}1 & 0 & 1 & 0 & 20\\ 1 & 1 & 0 & 0 & 80\\ 0 & \mathop{-}1 & 1 & 1 & 0\end{pmatrix} .

    REF = (10102001−1060000160)\begin{pmatrix}1 & 0 & 1 & 0 & 20\\ 0 & 1 & \mathop{-}1 & 0 & 60\\ 0 & 0 & 0 & 1 & 60\end{pmatrix} .

    {x1=20−x3x2=60+x3x4=60\begin{cases} x_1 = 20 - x_3 \\ x_2 = 60 + x_3 \\ x_4 = 60 \end{cases} , x3x_3 自由变量，x3≤20x_3 \le 20 .

  8. {T1=14(10+20+T2+T4)T2=14(20+40+T1+T3)T3=14(40+30+T2+T4)T4=14(10+30+T1+T3)\begin{cases}T_1 = \frac{1}{4} (10 + 20 + T_2 + T_4) \\ T_2 = \frac{1}{4} (20 + 40 + T_1 + T_3) \\ T_3 = \frac{1}{4}(40 + 30 + T_2 + T_4 ) \\ T_4 = \frac{1}{4} (10 + 30 + T_1 + T_3) \end{cases} .

    Aug. = (4−10−130−14−10600−14−170−10−1440)\begin{pmatrix}4 & \mathop{-}1 & 0 & \mathop{-}1 & 30\\ \mathop{-}1 & 4 & \mathop{-}1 & 0 & 60\\ 0 & \mathop{-}1 & 4 & \mathop{-}1 & 70\\ \mathop{-}1 & 0 & \mathop{-}1 & 4 & 40\end{pmatrix} .

    REF = (1−140−1415201−415−11518001−2716570001452)\begin{pmatrix}1 & - \frac{1}{4} & 0 & - \frac{1}{4} & \frac{15}{2}\\ 0 & 1 & - \frac{4}{15} & - \frac{1}{15} & 18\\ 0 & 0 & 1 & - \frac{2}{7} & \frac{165}{7}\\ 0 & 0 & 0 & 1 & \frac{45}{2}\end{pmatrix} .

    RREF = (10002001005520010300001452)\begin{pmatrix}1 & 0 & 0 & 0 & 20\\ 0 & 1 & 0 & 0 & \frac{55}{2}\\ 0 & 0 & 1 & 0 & 30\\ 0 & 0 & 0 & 1 & \frac{45}{2}\end{pmatrix} .

    (T1,T2,T3,T4)=(20,27.5,30,22.5)(T_1, T_2, T_3, T_4) = (20, 27.5, 30, 22.5) .

  9. 求和 AA 的所有行。

    (n,n,n,⋯ ,n,n)→(1,⋯ ,1)(n, n, n, \cdots, n, n) \to (1, \cdots, 1) .

    分别减去 AA 的所有行。所有变量的解均为 11 .

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