Question
Given $A=\left[\begin{array}{cc}2 & -6 \\ 2 & 0\end{array}\right] B=\left[\begin{array}{cc}-3 & 2 \\ 4 & 0\end{array}\right], C=\left[\begin{array}{ll}4 & 0 \\ 0 & 2\end{array}\right]$ Find the martix $X$ such that $A + 2X = 2B + C.$
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Answer
$\begin{array}{l} A =\left[\begin{array}{cc}2 & -6 \\ 2 & 0\end{array}\right] B =\left[\begin{array}{cc}-3 & 2 \\ 4 & 0\end{array}\right], C =\left[\begin{array}{ll}4 & 0 \\ 0 & 2\end{array}\right] \end{array} $
$ A +2 x =2 B + C $
$ {\left[\begin{array}{cc}2 & -6 \\ 2 & 0\end{array}\right]+2 X =2\left[\begin{array}{cc}-3 & 2 \\ 4 & 0\end{array}\right]+\left[\begin{array}{ll}4 & 0 \\ 0 & 2\end{array}\right]} $
$ 2 X =\left[\begin{array}{cc}-6 & 4 \\ 8 & 0\end{array}\right]+\left[\begin{array}{cc}4 & 0 \\ 0 & 2\end{array}\right]-\left[\begin{array}{cc}2 & -6 \\ 2 & 0\end{array}\right] $
$ 2 X =\left[\begin{array}{cc}-6+4-24+0+6 \\ 8+0-20+2-0\end{array}\right]=\left[\begin{array}{cc}-4 & 10 \\ 6 & 2\end{array}\right] $
$ 2 X =2\left[\begin{array}{cc}-2 & 5 \\ 3 & 1\end{array}\right] X =\left[\begin{array}{cc}-2 & 5 \\ 3 & 1\end{array}\right]$
$ A +2 x =2 B + C $
$ {\left[\begin{array}{cc}2 & -6 \\ 2 & 0\end{array}\right]+2 X =2\left[\begin{array}{cc}-3 & 2 \\ 4 & 0\end{array}\right]+\left[\begin{array}{ll}4 & 0 \\ 0 & 2\end{array}\right]} $
$ 2 X =\left[\begin{array}{cc}-6 & 4 \\ 8 & 0\end{array}\right]+\left[\begin{array}{cc}4 & 0 \\ 0 & 2\end{array}\right]-\left[\begin{array}{cc}2 & -6 \\ 2 & 0\end{array}\right] $
$ 2 X =\left[\begin{array}{cc}-6+4-24+0+6 \\ 8+0-20+2-0\end{array}\right]=\left[\begin{array}{cc}-4 & 10 \\ 6 & 2\end{array}\right] $
$ 2 X =2\left[\begin{array}{cc}-2 & 5 \\ 3 & 1\end{array}\right] X =\left[\begin{array}{cc}-2 & 5 \\ 3 & 1\end{array}\right]$
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