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 matrix $X$ such that $A+2 X=2 B+C$.
✓
Answer
$A + 2X = 2B + C$
$\begin{array}{l}\Rightarrow\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] \end{array} $
$ \Rightarrow\left[\begin{array}{cc}2 & -6 \\ 2 & 0\end{array}\right]+2 X=\left[\begin{array}{cc}-6+4 & 4+0 \\ 8+0 & 0+2\end{array}\right]$
$ \Rightarrow\left[\begin{array}{cc}2 & -6 \\ 2 & 0\end{array}\right]+2 X=\left[\begin{array}{cc}-2 & 4 \\ 8 & 2\end{array}\right]$
$\Rightarrow 2 X=\left[\begin{array}{cc}-2 & 4 \\ 8 & 2\end{array}\right]-\left[\begin{array}{cc}2 & -6 \\ 2 & 0\end{array}\right]=\left[\begin{array}{cc}-4 & 10 \\ 6 & 2\end{array}\right] $
$ \Rightarrow X=\frac{1}{2}\left[\begin{array}{cc}-4 & 10 \\ 6 & 2\end{array}\right] $
$ \Rightarrow X=\left[\begin{array}{cc}-2 & 5 \\ 3 & 1\end{array}\right]$
$\begin{array}{l}\Rightarrow\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] \end{array} $
$ \Rightarrow\left[\begin{array}{cc}2 & -6 \\ 2 & 0\end{array}\right]+2 X=\left[\begin{array}{cc}-6+4 & 4+0 \\ 8+0 & 0+2\end{array}\right]$
$ \Rightarrow\left[\begin{array}{cc}2 & -6 \\ 2 & 0\end{array}\right]+2 X=\left[\begin{array}{cc}-2 & 4 \\ 8 & 2\end{array}\right]$
$\Rightarrow 2 X=\left[\begin{array}{cc}-2 & 4 \\ 8 & 2\end{array}\right]-\left[\begin{array}{cc}2 & -6 \\ 2 & 0\end{array}\right]=\left[\begin{array}{cc}-4 & 10 \\ 6 & 2\end{array}\right] $
$ \Rightarrow X=\frac{1}{2}\left[\begin{array}{cc}-4 & 10 \\ 6 & 2\end{array}\right] $
$ \Rightarrow X=\left[\begin{array}{cc}-2 & 5 \\ 3 & 1\end{array}\right]$
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