Question
Find $X$ if $Y=\left[\begin{array}{ll}3 & 2 \\ 1 & 4\end{array}\right]$ and $2 X+Y=\left[\begin{array}{cc}1 & 0 \\ -3 & 2\end{array}\right]$
✓
Answer
Given
$
\begin{aligned}
& 2 X+Y=\left[\begin{array}{cc}
1 & 0 \\
-3 & 2
\end{array}\right] \\
& \Rightarrow 2 X=2 X+Y=\left[\begin{array}{cc}
1 & 0 \\
-3 & 2
\end{array}\right]-Y \\
& \Rightarrow 2 X=\left[\begin{array}{cc}
1 & 0 \\
-3 & 2
\end{array}\right]-\left[\begin{array}{ll}
3 & 2 \\
1 & 4
\end{array}\right] \\
& =\left[\begin{array}{cc}
1-3 & 0-2 \\
-3-1 & 2-4
\end{array}\right] \\
& =\left[\begin{array}{ll}
-2 & -2 \\
-4 & -2
\end{array}\right] \\
& X =\frac{1}{2}\left[\begin{array}{ll}
-2 & -2 \\
-4 & -2
\end{array}\right] \\
& =\left[\begin{array}{ll}
-1 & -1 \\
-2 & -1
\end{array}\right] . \\
&
\end{aligned}
$
$
\begin{aligned}
& 2 X+Y=\left[\begin{array}{cc}
1 & 0 \\
-3 & 2
\end{array}\right] \\
& \Rightarrow 2 X=2 X+Y=\left[\begin{array}{cc}
1 & 0 \\
-3 & 2
\end{array}\right]-Y \\
& \Rightarrow 2 X=\left[\begin{array}{cc}
1 & 0 \\
-3 & 2
\end{array}\right]-\left[\begin{array}{ll}
3 & 2 \\
1 & 4
\end{array}\right] \\
& =\left[\begin{array}{cc}
1-3 & 0-2 \\
-3-1 & 2-4
\end{array}\right] \\
& =\left[\begin{array}{ll}
-2 & -2 \\
-4 & -2
\end{array}\right] \\
& X =\frac{1}{2}\left[\begin{array}{ll}
-2 & -2 \\
-4 & -2
\end{array}\right] \\
& =\left[\begin{array}{ll}
-1 & -1 \\
-2 & -1
\end{array}\right] . \\
&
\end{aligned}
$
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