Дана матрица A. Найдите квадратную матрицу P, для которой матрица P A является улучшенным ступенчатым видом матрицы A, и выпишите этот вид.

$
A = \left(
\begin{array}{ccccc}
-2 & 1 & -2 & -4 & 8
\\
0 & 0 & 1 & 2 & -2
\\
-8 & 3 & -1 & -4 & 18
\\
5 & -2 & 1 & 3 & -12
\end{array}
\right)
=> \quad (3) = (3) - 4 * (1)
\left(
\begin{array}{ccccc}
-2 & 1 & -2 & -4 & 8
\\
0 & 0 & 1 & 2 & -2
\\
0 & -1 & 7 & 12 & -14
\\
5 & -2 & 1 & 3 & -12
\end{array}
\right)
=> \quad (4) = (4) + 2 * (1)
\\\\\\
\left(
\begin{array}{ccccc}
-2 & 1 & -2 & -4 & 8
\\
0 & 0 & 1 & 2 & -2
\\
0 & -1 & 7 & 12 & -14
\\
1 & 0 & -3 & -5 & 4
\end{array}
\right)
=> \quad (1) = (1) + 2 * (4)
\left(
\begin{array}{ccccc}
0 & 1 & -8 & -14 & 16
\\
0 & 0 & 1 & 2 & -2
\\
0 & -1 & 7 & 12 & -14
\\
1 & 0 & -3 & -5 & 4
\end{array}
\right)
=> \quad (1) <--> (4)
\\\\\\
\left(
\begin{array}{ccccc}
1 & 0 & -3 & -5 & 4
\\
0 & 0 & 1 & 2 & -2
\\
0 & -1 & 7 & 12 & -14
\\
0 & 1 & -8 & -14 & 16
\end{array}
\right)
=> \quad (2) <--> (4)
\left(
\begin{array}{ccccc}
1 & 0 & -3 & -5 & 4
\\
0 & 1 & -8 & -14 & 16
\\
0 & -1 & 7 & 12 & -14
\\
0 & 0 & 1 & 2 & -2
\end{array}
\right)
=> \quad (3) = (3) + (2)
\\\\\\
\left(
\begin{array}{ccccc}
1 & 0 & -3 & -5 & 4
\\
0 & 1 & -8 & -14 & 16
\\
0 & 0 & -1 & -2 & 2
\\
0 & 0 & 1 & 2 & -2
\end{array}
\right)
=> \quad (3) = (3) * (-1)
\left(
\begin{array}{ccccc}
1 & 0 & -3 & -5 & 4
\\
0 & 1 & -8 & -14 & 16
\\
0 & 0 & 1 & 2 & -2
\\
0 & 0 & 1 & 2 & -2
\end{array}
\right)
=> (4) = (4) - (3)
\\\\\\
\left(
\begin{array}{ccccc}
1 & 0 & -3 & -5 & 4
\\
0 & 1 & -8 & -14 & 16
\\
0 & 0 & 1 & 2 & -2
\\
0 & 0 & 0 & 0 & 0
\end{array}
\right)
=> (2) = (2) + 8 * (3)
\left(
\begin{array}{ccccc}
1 & 0 & -3 & -5 & 4
\\
0 & 1 & 0 & 2 & 0
\\
0 & 0 & 1 & 2 & -2
\\
0 & 0 & 0 & 0 & 0
\end{array}
\right)
=> (1) = (1) + 3 * (3)
\left(
\begin{array}{ccccc}
1 & 0 & 0 & 1 & -2
\\
0 & 1 & 0 & 2 & 0
\\
0 & 0 & 1 & 2 & -2
\\
0 & 0 & 0 & 0 & 0
\end{array}
\right)
\\\\\\
S_{1} =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0
\\
0 & 1 & 0 & 0
\\
-4 & 0 & 1 & 0
\\
0 & 0 & 0 & 1
\end{array}
\right)
\\\\\\
S_{2} =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0
\\
0 & 1 & 0 & 0
\\
0 & 0 & 1 & 0
\\
2 & 0 & 0 & 1
\end{array}
\right)
\\\\\\
S_{3} =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 2
\\
0 & 1 & 0 & 0
\\
0 & 0 & 1 & 0
\\
0 & 0 & 0 & 1
\end{array}
\right)
\\\\\\
S_{4} =
\left(
\begin{array}{cccc}
0 & 0 & 0 & 1
\\
0 & 1 & 0 & 0
\\
0 & 0 & 1 & 0
\\
1 & 0 & 0 & 0
\end{array}
\right)
\\\\\\
S_{5} =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0
\\
0 & 0 & 0 & 1
\\
0 & 0 & 1 & 0
\\
0 & 1 & 0 & 0
\end{array}
\right)
\\\\\\
S_{6} =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0
\\
0 & 1 & 0 & 0
\\
0 & 1 & 1 & 0
\\
0 & 0 & 0 & 1
\end{array}
\right)
\\\\\\
S_{7} =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0
\\
0 & 1 & 0 & 0
\\
0 & 0 & -1 & 0
\\
0 & 0 & 0 & 1
\end{array}
\right)
\\\\\\
S_{8} =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0
\\
0 & 1 & 0 & 0
\\
0 & 0 & 1 & 0
\\
0 & 0 & -1 & 1
\end{array}
\right)
\\\\\\
S_{9} =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0
\\
0 & 1 & 8 & 0
\\
0 & 0 & 1 & 0
\\
0 & 0 & 0 & 1
\end{array}
\right)
\\\\\\
S_{10} =
\left(
\begin{array}{cccc}
1 & 0 & 3 & 0
\\
0 & 1 & 0 & 0
\\
0 & 0 & 1 & 0
\\
0 & 0 & 0 & 1
\end{array}
\right)
\\\\\\
PA = S_{10} * S_{9} * S_{8} * S_{7} * S_{6} * S_{6} * S_{5} * S_{4} * S_{3} * S_{2} * S_{1} * A
\\\\\\
\left(
\begin{array}{cccc}
1 & 0 & 3 & 0
\\
0 & 1 & 0 & 0
\\
0 & 0 & 1 & 0
\\
0 & 0 & 0 & 1
\end{array}
\right)
*
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0
\\
0 & 1 & 8 & 0
\\
0 & 0 & 1 & 0
\\
0 & 0 & 0 & 1
\end{array}
\right)
=
\left(
\begin{array}{cccc}
1 & 0 & 3 & 0
\\
0 & 1 & 8 & 0
\\
0 & 0 & 1 & 0
\\
0 & 0 & 0 & 1
\end{array}
\right)
\\\\\\
\left(
\begin{array}{cccc}
1 & 0 & 3 & 0
\\
0 & 1 & 8 & 0
\\
0 & 0 & 1 & 0
\\
0 & 0 & 0 & 1
\end{array}
\right)
*
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0
\\
0 & 1 & 0 & 0
\\
0 & 0 & 1 & 0
\\
0 & 0 & -1 & 1
\end{array}
\right)
=
\left(
\begin{array}{cccc}
1 & 0 & 3 & 0
\\
0 & 1 & 8 & 0
\\
0 & 0 & 1 & 0
\\
0 & 0 & -1 & 1
\end{array}
\right)
\\\\\\
\left(
\begin{array}{cccc}
1 & 0 & 3 & 0
\\
0 & 1 & 8 & 0
\\
0 & 0 & 1 & 0
\\
0 & 0 & -1 & 1
\end{array}
\right)
*
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0
\\
0 & 1 & 0 & 0
\\
0 & 0 & -1 & 0
\\
0 & 0 & 0 & 1
\end{array}
\right)
=
\left(
\begin{array}{cccc}
1 & 0 & -3 & 0
\\
0 & 1 & -8 & 0
\\
0 & 0 & -1 & 0
\\
0 & 0 & 1 & 1
\end{array}
\right)
\\\\\\
\left(
\begin{array}{cccc}
1 & 0 & -3 & 0
\\
0 & 1 & -8 & 0
\\
0 & 0 & -1 & 0
\\
0 & 0 & 1 & 1
\end{array}
\right)
*
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0
\\
0 & 1 & 0 & 0
\\
0 & 1 & 1 & 0
\\
0 & 0 & 0 & 1
\end{array}
\right)
=
\left(
\begin{array}{cccc}
1 & -3 & -3 & 0
\\
0 & -7 & -8 & 0
\\
0 & -1 & -1 & 0
\\
0 & 1 & 1 & 1
\end{array}
\right)
\\\\\\
\left(
\begin{array}{cccc}
1 & -3 & -3 & 0
\\
0 & -7 & -8 & 0
\\
0 & -1 & -1 & 0
\\
0 & 1 & 1 & 1
\end{array}
\right)
*
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0
\\
0 & 0 & 0 & 1
\\
0 & 0 & 1 & 0
\\
0 & 1 & 0 & 0
\end{array}
\right)
=
\left(
\begin{array}{cccc}
1 & 0 & -3 & -3
\\
0 & 0 & -8 & -7
\\
0 & 0 & -1 & -1
\\
0 & 1 & 1 & 1
\end{array}
\right)
\\\\\\
\left(
\begin{array}{cccc}
1 & 0 & -3 & -3
\\
0 & 0 & -8 & -7
\\
0 & 0 & -1 & -1
\\
0 & 1 & 1 & 1
\end{array}
\right)
*
\left(
\begin{array}{cccc}
0 & 0 & 0 & 1
\\
0 & 1 & 0 & 0
\\
0 & 0 & 1 & 0
\\
1 & 0 & 0 & 0
\end{array}
\right)
=
\left(
\begin{array}{cccc}
-3 & 0 & -3 & 1
\\
-7 & 0 & -8 & 0
\\
-1 & 0 & -1 & 0
\\
1 & 1 & 1 & 0
\end{array}
\right)
\\\\\\
\left(
\begin{array}{cccc}
-3 & 0 & -3 & 1
\\
-7 & 0 & -8 & 0
\\
-1 & 0 & -1 & 0
\\
1 & 1 & 1 & 0
\end{array}
\right)
*
\left(
\begin{array}{cccc}
1 & 0 & 0 & 2
\\
0 & 1 & 0 & 0
\\
0 & 0 & 1 & 0
\\
0 & 0 & 0 & 1
\end{array}
\right)
=
\left(
\begin{array}{cccc}
-3 & 0 & -3 & -5
\\
-7 & 0 & -8 & -14
\\
-1 & 0 & -1 & -2
\\
1 & 1 & 1 & 2
\end{array}
\right)
\\\\\\
\left(
\begin{array}{cccc}
-3 & 0 & -3 & -5
\\
-7 & 0 & -8 & -14
\\
-1 & 0 & -1 & -2
\\
1 & 1 & 1 & 2
\end{array}
\right)
*
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0
\\
0 & 1 & 0 & 0
\\
0 & 0 & 1 & 0
\\
2 & 0 & 0 & 1
\end{array}
\right)
=
\left(
\begin{array}{cccc}
-13 & 0 & -3 & -5
\\
-35 & 0 & -8 & -14
\\
-5 & 0 & -1 & -2
\\
5 & 1 & 1 & 2
\end{array}
\right)
\\\\\\
\left(
\begin{array}{cccc}
-13 & 0 & -3 & -5
\\
-35 & 0 & -8 & -14
\\
-5 & 0 & -1 & -2
\\
5 & 1 & 1 & 2
\end{array}
\right)
*
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0
\\
0 & 1 & 0 & 0
\\
-4 & 0 & 1 & 0
\\
0 & 0 & 0 & 1
\end{array}
\right)
=
\left(
\begin{array}{cccc}
-1 & 0 & -3 & -5
\\
-3 & 0 & -8 & -14
\\
-1 & 0 & -1 & -2
\\
1 & 1 & 1 & 2
\end{array}
\right)
\\\\\\
\text{И наконец } PA =
\\\\\\
\left(
\begin{array}{cccc}
-1 & 0 & -3 & -5
\\
-3 & 0 & -8 & -14
\\
-1 & 0 & -1 & -2
\\
1 & 1 & 1 & 2
\end{array}
\right)
*
\left(
\begin{array}{ccccc}
-2 & 1 & -2 & -4 & 8
\\
0 & 0 & 1 & 2 & -2
\\
-8 & 3 & -1 & -4 & 18
\\
5 & -2 & 1 & 3 & -12
\end{array}
\right)
=
\left(
\begin{array}{ccccc}
1 & 0 & 0 & 1 & -2
\\
0 & 1 & 0 & 2 & 0
\\
0 & 0 & 1 & 2 & -2
\\
0 & 0 & 0 & 0 & 0
\end{array}
\right)
\\\\\
\\\\\
\text{Ответ: }
\begin{equation*}
P = \left(
\begin{array}{cccc}
-1 & 0 & -3 & -5
\\
-3 & 0 & -8 & -14
\\
-1 & 0 & -1 & -2
\\
1 & 1 & 1 & 2
\end{array}
\right)
\end{equation*}
$