Даны матрицы A, B, C, D. Выясните, имеют ли системы $ ABC^{−1}x = 0 $ и Dx = 0, где x ∈ $ R^4 $ , одинаковое множество решений.

$ A = \left(
\begin{array}{ccc}
1 & 0 & 0
\\
0 & 1 & 0
\\
0 & 0 & 1
\\
6 & -7 & 7
\end{array}
\right)
\quad
B = \left(
\begin{array}{cccc}
1 & 0 & 4 & 5
\\
0 & 1 & -3 & 2
\\
1 & 2 & -2 & 9
\end{array}
\right)
\quad
C = \left(
\begin{array}{cccc}
1 & -1 & -2 & 1
\\
2 & -1 & -4 & 0
\\
-1 & 3 & 3 & -4
\\
1 & -1 & -2 & 2
\end{array}
\right)
D = \left(
\begin{array}{cccc}
29 & -11 & 6 & 0
\\
-19 & 3 & 0 & 14
\\
86 & -20 & 6 & -42
\\
49 & -27 & 18 & 28
\end{array}
\right)
\\\\\\
A * B = \left(
\begin{array}{ccc}
1 & 0 & 0
\\
0 & 1 & 0
\\
0 & 0 & 1
\\
6 & -7 & 7
\end{array}
\right)
*
\left(
\begin{array}{cccc}
1 & 0 & 4 & 5
\\
0 & 1 & -3 & 2
\\
1 & 2 & -2 & 9
\end{array}
\right)
=
\left(
\begin{array}{cccc}
1 & 0 & 4 & 5
\\
0 & 1 & -3 & 2
\\
1 & 2 & -2 & 9
\\
13 & 7 & 31 & 79
\end{array}
\right)
\\\\\\
A * B * C^{-1} = \left(
\begin{array}{cccc}
1 & 0 & 4 & 5
\\
0 & 1 & -3 & 2
\\
1 & 2 & -2 & 9
\\
13 & 7 & 31 & 79
\end{array}
\right)
*
\left(
\begin{array}{cccc}
1 & 2 & -1 & 1
\\
-1 & -1 & 3 & -1
\\
-2 & -4 & 3 & -2
\\
1 & 0 & -4 & 2
\end{array}
\right)
=
\left(
\begin{array}{cccc}
-2 & -14 & -9 & 3
\\
7 & 11 & -14 & 9
\\
12 & 8 & -37 & 21
\\
23 & -105 & -215 & 102
\end{array}
\right)
\\\\\\
A * B * C^{-1} * X = \left(
\begin{array}{cccc}
-2 & -14 & -9 & 3
\\
7 & 11 & -14 & 9
\\
12 & 8 & -37 & 21
\\
23 & -105 & -215 & 102
\end{array}
\right)
*
\left(
\begin{array}{c}
x_{1}
\\
x_{2}
\\
x_{3}
\\
x_{4}
\end{array}
\right)
=
\left(
\begin{array}{cccc}
-2 * x_{1} - 14 * x_{2} - 9 * x_{3} + 3 * x_{4}
\\
7 * x_{1} + 11 * x_{2} - 14 * x_{3} + 9 * x_{4}
\\
12 * x_{1} + 8 * x_{2} -37 * x_{3} + 21 * x_{4}
\\
23 * x_{1} - 105 * x_{2} - 215* x_{3} + 102 * x_{4}
\end{array}
\right)
\\\\\\
D * X = \left(
\begin{array}{cccc}
29 & -11 & 6 & 0
\\
-19 & 3 & 0 & 14
\\
86 & -20 & 6 & -42
\\
49 & -27 & 18 & 28
\end{array}
\right)
*
\left(
\begin{array}{c}
x_{1}
\\
x_{2}
\\
x_{3}
\\
x_{4}
\end{array}
\right)
=
\left(
\begin{array}{cccc}
-29 * x_{1} - 11 * x_{2} + 6 * x_{3}
\\
-19 * x_{1} + 3 * x_{2} + 14 * x_{4}
\\
86 * x_{1} - 20 * x_{2} + 6 * x_{3} - 42 * x_{4}
\\
49 * x_{1} - 27 * x_{2} + 18 * x_{3} + 28 * x_{4}
\end{array}
\right)
\\\\\\
\text{Приведем к ступенчатому виду: } A * B * C^{-1} * X
\\\\\\
\left(
\begin{array}{cccc|c}
-2 & -14 & -9 & 3 & 0
\\
7 & 11 & -14 & 9 & 0
\\
12 & 8 & -37 & 21 & 0
\\
23 & -105 & -215 & 102 & 0
\end{array}
\right)
=> (4) = (4) + 11 * (1)
\left(
\begin{array}{cccc|c}
-2 & -14 & -9 & 3 & 0
\\
7 & 11 & -14 & 9 & 0
\\
12 & 8 & -37 & 21 & 0
\\
1 & -259 & -314 & 155 & 0
\end{array}
\right)
\\\\\\
=> (1) = \frac{(1)}{2}
\left(
\begin{array}{cccc|c}
1 & 7 & \frac{9}{2} & \frac{-3}{2} & 0
\\
7 & 11 & -14 & 9 & 0
\\
12 & 8 & -37 & 21 & 0
\\
1 & -259 & -314 & 155 & 0
\end{array}
\right)
=> (2) = (2) - 7 * (1)
\left(
\begin{array}{cccc|c}
1 & 7 & \frac{9}{2} & \frac{-3}{2} & 0
\\
\\
0 & -38 & \frac{-91}{2} & \frac{39}{2} & 0
\\
\\
12 & 8 & -37 & 21 & 0
\\
\\
1 & -259 & -314 & 155 & 0
\end{array}
\right)
\\\\\\
=> (3) = (3) -12 * (1)
\left(
\begin{array}{cccc|c}
1 & 7 & \frac{9}{2} & \frac{-3}{2} & 0
\\
\\
0 & -38 & \frac{-91}{2} & \frac{39}{2} & 0
\\
\\
0 & -76 & -91 & 39 & 0
\\
\\
1 & -259 & -314 & 155 & 0
\end{array}
\right)
=> (4) = (4) - (1)
\left(
\begin{array}{cccc|c}
1 & 7 & \frac{9}{2} & \frac{-3}{2} & 0
\\
\\
0 & -38 & \frac{-91}{2} & \frac{39}{2} & 0
\\
\\
0 & -76 & -91 & 39 & 0
\\
\\
0 & -266 & -\frac{637}{2} & \frac{313}{2} & 0
\end{array}
\right)
\\\\\\
=> (2) = -\frac{(2)}{38}
\left(
\begin{array}{cccc|c}
1 & 7 & \frac{9}{2} & \frac{-3}{2} & 0
\\
\\
0 & 1 & \frac{91}{76} & \frac{-39}{76} & 0
\\
\\
0 & -76 & -91 & 39 & 0
\\
\\
0 & -266 & -\frac{637}{2} & \frac{313}{2} & 0
\end{array}
\right)
=> (1) = (1) - 7 * (2)
\left(
\begin{array}{cccc|c}
1 & 0 & -\frac{295}{76} & \frac{159}{76} & 0
\\
\\
0 & 1 & \frac{91}{76} & \frac{-39}{76} & 0
\\
\\
0 & -76 & -91 & 39 & 0
\\
\\
0 & -266 & -\frac{637}{2} & \frac{313}{2} & 0
\end{array}
\right)
\\\\\\
=> (3) = (3) + 76 * (2)
\left(
\begin{array}{cccc|c}
1 & 0 & -\frac{295}{76} & \frac{159}{76} & 0
\\
\\
0 & 1 & \frac{91}{76} & \frac{-39}{76} & 0
\\
\\
0 & 0 & 0 & 0 & 0
\\
\\
0 & -266 & -\frac{637}{2} & \frac{313}{2} & 0
\end{array}
\right)
=> (4) = (4) + 266 * (2)
\left(
\begin{array}{cccc|c}
1 & 0 & -\frac{295}{76} & \frac{159}{76} & 0
\\
\\
0 & 1 & \frac{91}{76} & \frac{-39}{76} & 0
\\
\\
0 & 0 & 0 & 0 & 0
\\
\\
0 & 0 & 0 & 20 & 0
\end{array}
\right)
\\\\\\
=> (4) <--> (3)
\left(
\begin{array}{cccc|c}
1 & 0 & -\frac{295}{76} & \frac{159}{76} & 0
\\
\\
0 & 1 & \frac{91}{76} & \frac{-39}{76} & 0
\\
\\
0 & 0 & 0 & 20 & 0
\\
\\
0 & 0 & 0 & 0 & 0
\end{array}
\right)
=> (3) = \frac{(3)}{20}
\left(
\begin{array}{cccc|c}
1 & 0 & -\frac{295}{76} & \frac{159}{76} & 0
\\
\\
0 & 1 & \frac{91}{76} & \frac{-39}{76} & 0
\\
\\
0 & 0 & 0 & 1 & 0
\\
\\
0 & 0 & 0 & 0 & 0
\end{array}
\right)
\\\\\\
=> (1) = (1) - \frac{159}{76} * (3)
\left(
\begin{array}{cccc|c}
1 & 0 & -\frac{295}{76} & 0 & 0
\\
\\
0 & 1 & \frac{91}{76} & \frac{-39}{76} & 0
\\
\\
0 & 0 & 0 & 1 & 0
\\
\\
0 & 0 & 0 & 0 & 0
\end{array}
\right)
=> (2) = (2) + \frac{39}{76} * (3)
\left(
\begin{array}{cccc|c}
1 & 0 & -\frac{295}{76} & 0 & 0
\\
\\
0 & 1 & \frac{91}{76} & 0 & 0
\\
\\
0 & 0 & 0 & 1 & 0
\\
\\
0 & 0 & 0 & 0 & 0
\end{array}
\right)
\\\\\\
\text{Результат: }
\left(
\begin{array}{c}
x_{1}
\\
x_{2}
\\
x_{3}
\\
x_{4}
\end{array}
\right)
=
\left(
\begin{array}{cccc|c}
\frac{295}{76}x_{3}
\\
\\
-\frac{91}{76}x_{3}
\\
\\
x_{3}
\\
\\
0
\end{array}
\right), x_{3} \in \mathbb{R}
\\\\\\
\\\\\\
\\\\\\
\text{Приведем к ступенчатому виду: } D * X
\\\\\\
\left(
\begin{array}{ccccc}
29 & -11 & 6 & 0 & 0
\\
-19 & 3 & 0 & 14 & 0
\\
86 & -20 & 6 & -42 & 0
\\
49 & -27 & 18 & 28 & 0
\end{array}
\right)
(1) = \frac{(1)}{29}
\left(
\begin{array}{ccccc}
1 & \frac{-11}{29} & \frac{6}{29} & 0 & 0
\\
-19 & 3 & 0 & 14 & 0
\\
86 & -20 & 6 & -42 & 0
\\
49 & -27 & 18 & 28 & 0
\end{array}
\right)
(2) = (2) + 19 * (1)
\left(
\begin{array}{ccccc}
1 & \frac{-11}{29} & \frac{6}{29} & 0 & 0
\\
\\
0 & \frac{-122}{29} & \frac{114}{29} & 14 & 0
\\
\\
86 & -20 & 6 & -42 & 0
\\
\\
49 & -27 & 18 & 28 & 0
\end{array}
\right)
\\\\\\
(3) = (3) - 86 * (1)
\left(
\begin{array}{ccccc}
1 & \frac{-11}{29} & \frac{6}{29} & 0 & 0
\\
\\
0 & \frac{-122}{29} & \frac{114}{29} & 14 & 0
\\
\\
0 & \frac{366}{29} & \frac{-342}{29} & -42 & 0
\\
\\
49 & -27 & 18 & 28 & 0
\end{array}
\right)
(4) = (4) - 49 * (1)
\left(
\begin{array}{ccccc}
1 & \frac{-11}{29} & \frac{6}{29} & 0 & 0
\\
\\
0 & \frac{-122}{29} & \frac{114}{29} & 14 & 0
\\
\\
0 & \frac{366}{29} & \frac{-342}{29} & -42 & 0
\\
\\
0 & \frac{-244}{29} & \frac{228}{29} & 28 & 0
\end{array}
\right)
\\\\\\
(2) = \frac{-29}{122} * (2)
\left(
\begin{array}{ccccc}
1 & \frac{-11}{29} & \frac{6}{29} & 0 & 0
\\
\\
0 & 1 & \frac{-57}{61} & -\frac{203}{61} & 0
\\
\\
0 & \frac{366}{29} & \frac{-342}{29} & -42 & 0
\\
\\
0 & \frac{-244}{29} & \frac{228}{29} & 28 & 0
\end{array}
\right)
(1) = (1) + \frac{11}{29} * (2)
\left(
\begin{array}{ccccc}
1 & 0 & \frac{-9}{61} & \frac{-77}{61} & 0
\\
\\
0 & 1 & \frac{-57}{61} & -\frac{203}{61} & 0
\\
\\
0 & \frac{366}{29} & \frac{-342}{29} & -42 & 0
\\
\\
0 & \frac{-244}{29} & \frac{228}{29} & 28 & 0
\end{array}
\right)
\\\\\\
(3) = (3) - \frac{366}{29} * (2)
\left(
\begin{array}{ccccc}
1 & 0 & \frac{-9}{61} & \frac{-77}{61} & 0
\\
\\
0 & 1 & \frac{-57}{61} & -\frac{203}{61} & 0
\\
\\
0 & 0 & 0 & 0 & 0
\\
\\
0 & \frac{-244}{29} & \frac{228}{29} & 28 & 0
\end{array}
\right)
(4) = (4) + \frac{244}{29} * (2)
\left(
\begin{array}{ccccc}
1 & 0 & \frac{-9}{61} & \frac{-77}{61} & 0
\\
\\
0 & 1 & \frac{-57}{61} & -\frac{203}{61} & 0
\\
\\
0 & 0 & 0 & 0 & 0
\\
\\
0 & 0 & 0 & 0 & 0
\end{array}
\right)
\\\\\\
\text{Результат: }
\left(
\begin{array}{c}
x_{1}
\\
x_{2}
\\
x_{3}
\\
x_{4}
\end{array}
\right)
=
\left(
\begin{array}{cccc|c}
\frac{9}{61}x_{3} + \frac{77}{61}x_{4}
\\
\\
\frac{57}{61}x_{3} + \frac{203}{61}x_{4}
\\
\\
x_{3}
\\
\\
x_{4}
\end{array}
\right), x_{3}, x_{4} \in \mathbb{R}
\\\\\\
\text{Из решения систем, заметим, что системы не имеют одинаковое множество решений.}
$