линейная-алгебра - Как решать матричное уравнение?

Предполагаю, что задачу можно решить следующим образом:

$% 1. \ A = \begin {bmatrix} 1&-1 \\ 0&0 \end {bmatrix} \wedge B = \begin {bmatrix} -3&2 \\ 1&4 \end {bmatrix} \wedge X = \begin {bmatrix} x_{11}&x_{12} \\ x_{21}&x_{22} \end {bmatrix} \wedge 2X - B = A \ast (X + 2B) $%

$% \Rightarrow 2 \cdot \begin {bmatrix} x_{11}&x_{12} \\ x_{21}&x_{22} \end {bmatrix} - \begin {bmatrix} -3&2 \\ 1&4 \end {bmatrix} = \begin {bmatrix} 1&-1 \\ 0&0 \end {bmatrix} \ast (\begin {bmatrix} x_{11}&x_{12} \\ x_{21}&x_{22} \end {bmatrix} + 2 \cdot \begin {bmatrix} -3&2 \\ 1&4 \end {bmatrix})$%

$% \Leftrightarrow \begin {bmatrix} 2x_{11}&2x_{12} \\ 2x_{21}&2x_{22} \end {bmatrix} - \begin {bmatrix} -3&2 \\ 1&4 \end {bmatrix} = \begin {bmatrix} 1&-1 \\ 0&0 \end {bmatrix} \ast (\begin {bmatrix} x_{11}&x_{12} \\ x_{21}&x_{22} \end {bmatrix} + \begin {bmatrix} -6&4 \\ 2&8 \end {bmatrix})$%

$% \Leftrightarrow \begin {bmatrix} 2x_{11}+3&2x_{12}-2 \\ 2x_{21}-1&2x_{22}-4 \end {bmatrix} = \begin {bmatrix} 1&-1 \\ 0&0 \end {bmatrix} \ast \begin {bmatrix} x_{11}-6&x_{12}+4 \\ x_{21}+2&x_{22}+8 \end {bmatrix}$%

$% \Leftrightarrow \begin {bmatrix} 2x_{11}+3&2x_{12}-2 \\ 2x_{21}-1&2x_{22}-4 \end {bmatrix} = \begin {bmatrix} \langle 1,-1 \rangle \cdot \langle x_{11}-6, x_{21}+2 \rangle& \langle 1,-1 \rangle \cdot \langle x_{12}+4, x_{22}+8 \rangle \\ \langle 0,0 \rangle \cdot \langle x_{11}-6, x_{21}+2 \rangle & \langle 0,0 \rangle \cdot \langle x_{12}+4, x_{22}+8 \rangle \end {bmatrix} $%

$% \Leftrightarrow \begin {bmatrix} 2x_{11}+3&2x_{12}-2 \\ 2x_{21}-1&2x_{22}-4 \end {bmatrix} = \begin {bmatrix} 1 \cdot (x_{11}-6) + (-1) \cdot (x_{21}+2) & 1 \cdot (x_{12}+4) + (-1) \cdot (x_{22}+8) \\ 0 \cdot (x_{11}-6) + 0 \cdot (x_{21}+2) & 0 \cdot (x_{12}+4) + 0 \cdot (x_{22}+8) \end {bmatrix} $%

$% \Leftrightarrow \begin {bmatrix} 2x_{11}+3&2x_{12}-2 \\ 2x_{21}-1&2x_{22}-4 \end {bmatrix} = \begin {bmatrix} x_{11} - x_{21} - 8 & x_{12} - x_{22} - 4 \\ 0 & 0 \end {bmatrix} $%

$% \Rightarrow \begin {cases} 2x_{11}+3 = x_{11} - x_{21} - 8 \wedge 2x_{12}-2 = x_{12} - x_{22} - 4 \\ 2x_{21}-1 = 0 \wedge 2x_{22}-4 = 0 \end {cases} $%

$% \Leftrightarrow \begin {cases} x_{11} = - x_{21} - 11 \wedge x_{12} = - x_{22} - 2 \\ x_{21} = \frac{1}{2} \wedge x_{22} = 2 \end {cases} \Leftrightarrow \begin {cases} x_{11} = - 11\frac{1}{2} \wedge x_{12} = - 4 \\ x_{21} = \frac{1}{2} \wedge x_{22} = 2 \end {cases}$%

$%2. \ \begin {cases} x_{11} = -11.5 \wedge x_{12} = - 4.0 \\ x_{21} = 0.5 \wedge x_{22} = 2.0 \end {cases} \wedge X = \begin {bmatrix} x_{11}&x_{12} \\ x_{21}&x_{22} \end {bmatrix} \Rightarrow X = \begin {bmatrix} -11.5&-4.0 \\ 0.5&2.0 \end {bmatrix}$%