Дана матрица A. Найдите все матрицы X удовлетворяющие условию AX = XA.

Из условия: $ \, \, A = \left(
\begin{array}{ccc}
0 & 0 & 0
\\
6 & 7 & 4
\\
-2 & 0 & 8
\end{array}
\right)
\quad
X = \left(
\begin{array}{ccc}
* & 0 & 0
\\
* & * & *
\\
* & 0 & *
\end{array}
\right)
AX = XA
\left(
\begin{array}{ccc}
0 & 0 & 0
\\
6 & 7 & 4
\\
-2 & 0 & 8
\end{array}
\right)
*
\left(
\begin{array}{ccc}
x_{1} & 0 & 0
\\
x_{2} & x_{3} & x_{4}
\\
x_{5} & 0 & x_{6}
\end{array}
\right)
=
\left(
\begin{array}{ccc}
x_{1} & 0 & 0
\\
x_{2} & x_{3} & x_{4}
\\
x_{5} & 0 & x_{6}
\end{array}
\right)
*
\left(
\begin{array}{ccc}
0 & 0 & 0
\\
6 & 7 & 4
\\
-2 & 0 & 8
\end{array}
\right)

(**) \\
\\
\left(
\begin{array}{ccc}
0 & 0 & 0
\\
6 * x_{1} + 7 * x_{2} + 4 * x_{5} & 7 * x_{3} & 7 * x_{4} + 4 * x_{6}
\\
-2 * x_{1} + 8 * x_{5} & 0 & 8 * x_{6}
\end{array}
\right)
=
\left(
\begin{array}{ccc}
0 & 0 & 0
\\
6 * x_{3} - 2 * x_{4} & 7 * x_{3} & 4 * x_{3} + 8 * x_{4}
\\
-2 * x_{6} & 0 & 8 * x_{6}
\end{array}
\right)

\\ \\ \\

\begin{cases}
6 * x_{1} + 7 * x_{2} + 4 * x_{5} = 6 * x_{3} - 2 * x_{4}
\\
7 * x_{3} = 7 * x_{3}
\\
7 * x_{4} + 4 * x_{6} = 4 * x_{3} + 8 * x_{4}
\\
-2 * x_{1} + 8 * x_{5} = -2 * x_{6}
\\
8 * x_{6} = 8 * x_{6}
\end{cases}
=> \,
\begin{cases}
x_{1} = \frac{6 * x_{3} - 2 * x_{4} - 4 * x_{5} - 7 * x_{2}}{6}
\\
x_{4} = 4 * x_{6} - 4 * x_{3}
\\
x_{5} = \frac{-x_{6} + x_{1}}{4}
\end{cases}
=> \,
\\\\\\
\begin{cases}
x_{1} = \frac{6 * x_{3} - 2 * x_{4} - 4 * x_{5} - 7 * x_{2}}{6}
\\
x_{4} = 4 * x_{6} - 4 * x_{3}
\\
x_{5} = \frac{-x_{6} + \frac{6 * x_{3} - 2 * (4 * x_{6} - 4 * x_{3}) - 4 * x_{5} - 7 * x_{2}}{6}}{4}
\end{cases}
=> \,
\begin{cases}
x_{1} = \frac{6 * x_{3} - 2 * x_{4} - 4 * x_{5} - 7 * x_{2}}{6}
\\
x_{4} = 4 * x_{6} - 4 * x_{3}
\\
x_{5} = \frac{-14 * x_{6} - 14 * x_{3} - 4 * x_{5} - 7 * x_{2}}{24}
\end{cases}
=> \,
\begin{cases}
x_{1} = \frac{6 * x_{3} - 2 * x_{4} - 4 * x_{5} - 7 * x_{2}}{6}
\\
x_{4} = 4 * x_{6} - 4 * x_{3}
\\
x_{5} = \frac{-14 * x_{6} - 14 * x_{3} - 7 * x_{2}}{28}
\end{cases}
=> \,
\\\\\\
\begin{cases}
x_{1} = \frac{6 * x_{3} - 2 * x_{4} - 4 * x_{5} - 7 * x_{2}}{6}
\\
x_{4} = 4 * x_{6} - 4 * x_{3}
\\
x_{5} = \frac{-2 * x_{6} - 2 * x_{3} - x_{2}}{4}
\end{cases}
=> \,
\\\\\\
\begin{cases}
x_{1} = \frac{6 * x_{3} - 2 * x_{4} - 4 * \frac{-2 * x_{6} - 2 * x_{3} - x_{2}}{4} - 7 * x_{2}}{6}
\\
x_{4} = 4 * x_{6} - 4 * x_{3}
\\
x_{5} = \frac{-2 * x_{6} - 2 * x_{3} - x_{2}}{4}
\end{cases}
=> \,
\begin{cases}
x_{1} = \frac{6 * x_{3} - 2 * (4 * x_{6} - 4 * x_{3}) + 2 * x_{6} + 2 * x_{3} + x_{2} - 7 * x_{2}}{6}
\\
x_{4} = 4 * x_{6} - 4 * x_{3}
\\
x_{5} =\frac{-2 * x_{6} - 2 * x_{3} - x_{2}}{4}
\end{cases}
=> \,
\\\\\\
\begin{cases}
x_{1} = \frac{16 * x_{3} - 6 * x_{6} - 6 * x_{2}}{6}
\\
x_{4} = 4 * x_{6} - 4 * x_{3}
\\
x_{5} = \frac{-2 * x_{6} - 2 * x_{3} - x_{2}}{4}
\end{cases}
=> \,
\begin{cases}
x_{1} = \frac{8 * x_{3} + 3 * x_{6} - 3 * x_{2}}{3}
\\
x_{4} = 4 * x_{6} - 4 * x_{3}
\\
x_{5} = \frac{-2 * x_{6} - 2 * x_{3} - x_{2}}{4}
\end{cases}
$
Ответ:
\begin{equation*}
X = \left(
\begin{array}{ccc}
\frac{8 * x_{3} + 3 * x_{6} - 3 * x_{2}}{3} & 0 & 0
\\
x_{2} & x_{3} & 4 * x_{6} - 4 * x_{3}
\\
\frac{-2 * x_{6} - 2 * x_{3} - x_{2}}{4} & 0 & x_{6}
\end{array}
\right)
\text{, где} \quad x_{2}, x_{3}, x_{6} \in \mathbb{R}
\end{equation*}

Sanity check:
$ \begin{cases}
x_{1} = -4
\\
x_{2} = 4
\\
x_{3} = 0
\\
x_{4} = 0
\\
x_{5} = -1
\\
x_{6} = 0
\end{cases} $
=> Подставляем в (**) и получаем:
$
\left(
\begin{array}{ccc}
0 & 0 & 0
\\
0 & 0 & 0
\\
0 & 0 & 0
\end{array}
\right)
=
\left(
\begin{array}{ccc}
0 & 0 & 0
\\
0 & 0 & 0
\\
0 & 0 & 0
\end{array}
\right)
$