Given $A = \begin{bmatrix} 4 & -12 \\ 4 & 0 \end{bmatrix}$, $B = \begin{bmatrix} -3 & 2 \\ 4 & 1 \end{bmatrix}$ and $C = \begin{bmatrix} 4 & 0 \\ 0 & 0 \end{bmatrix}$, find matrix $X$ such that $A + 2X = 2B + C$
$A + 2X = 2B + C$
i.e. $\;$ $2X = 2B + C - A$
i.e. $\;$ $X = B + \dfrac{1}{2} \left(C - A\right)$
$\begin{aligned}
i.e. \;\; X & = \begin{bmatrix}
-3 & 2 \\
4 & 1
\end{bmatrix} + \dfrac{1}{2} \left\{\begin{bmatrix}
4 & 0 \\
0 & 0
\end{bmatrix} - \begin{bmatrix}
4 & -12 \\
4 & 0
\end{bmatrix} \right\} \\\\
& = \begin{bmatrix}
-3 & 2 \\
4 & 1
\end{bmatrix} + \dfrac{1}{2} \begin{bmatrix}
4 - 4 & 0 + 12 \\
0 - 4 & 0 - 0
\end{bmatrix} \\\\
& = \begin{bmatrix}
-3 & 2 \\
4 & 1
\end{bmatrix} + \dfrac{1}{2} \begin{bmatrix}
0 & 12 \\
-4 & 0
\end{bmatrix} \\\\
& = \begin{bmatrix}
-3 & 2 \\
4 & 1
\end{bmatrix} + \begin{bmatrix}
0 & 6 \\
-2 & 0
\end{bmatrix} \\\\
& = \begin{bmatrix}
-3 + 0 & 2 + 6 \\
4 - 2 & 1 + 0
\end{bmatrix} \\\\
& = \begin{bmatrix}
-3 & 8 \\
2 & 1
\end{bmatrix}
\end{aligned}$