Question

If $X+Y=\left[\begin{array}{cc}5& 2\\ 0& 9\end{array}\right]$ and $X-Y=\left[\begin{array}{cc}3& 6\\ 0& -1\end{array}\right],$ then

• A. $X=\left[\begin{array}{cc}4& 4\\ 0& 4\end{array}\right]$
• B. $X=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$
• C. $Y=\left[\begin{array}{cc}1& -2\\ 0& 5\end{array}\right]$
• D. $Y=\left[\begin{array}{cc}-2& 1\\ 0& 5\end{array}\right]$

Answer 1

Answer: (A), (C)

$Y=\left[\begin{array}{cc}1& -2\\ 0& 5\end{array}\right]$

$(X+Y)+(X-Y)=\left[\begin{array}{cc}5& 2\\ 0& 9\end{array}\right]+\left[\begin{array}{cc}3& 6\\ 0& -1\end{array}\right]$
$2X=\left[\begin{array}{cc}5+3& 2+6\\ 0+0& 9-1\end{array}\right]$
$2X=\left[\begin{array}{cc}8& 8\\ 0& 8\end{array}\right]$
$X=\frac{1}{2}\left[\begin{array}{cc}8& 8\\ 0& 8\end{array}\right]$
$X=\left[\begin{array}{cc}4& 4\\ 0& 4\end{array}\right]$
Now,
$(X+Y)-(X-Y)=\left[\begin{array}{cc}5& 2\\ 0& 9\end{array}\right]-\left[\begin{array}{cc}3& 6\\ 0& -1\end{array}\right]$
$2Y=\left[\begin{array}{cc}5-3& 2-6\\ 0-1& 9+1\end{array}\right]$
$2Y=\left[\begin{array}{cc}2& -4\\ 0& 10\end{array}\right]$
$Y=\frac{1}{2}\left[\begin{array}{cc}2& -4\\ 0& 10\end{array}\right]$
$Y=\left[\begin{array}{cc}1& -2\\ 0& 5\end{array}\right]$
Hence, we have $X=\left[\begin{array}{cc}4& 4\\ 0& 4\end{array}\right]$ and $Y=\left[\begin{array}{cc}1& -2\\ 0& 5\end{array}\right]$