Question

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

Answer 1

$$\begin{gathered} \mathrm{Given}:\left(X+Y\right)+\left(X-Y\right)=\left[\begin{array}{cc}5& 2\\ 0& 9\end{array}\right]+\left[\begin{array}{cc}3& 6\\ 0& -1\end{array}\right] \\ \Rightarrow 2X=\left[\begin{array}{cc}5+3& 2+6\\ 0+0& 9-1\end{array}\right] \\ \Rightarrow 2X=\left[\begin{array}{cc}8& 8\\ 0& 8\end{array}\right] \\ \Rightarrow X=\frac{1}{2}\left[\begin{array}{cc}8& 8\\ 0& 8\end{array}\right] \\ \Rightarrow X=\left[\begin{array}{cc}4& 4\\ 0& 4\end{array}\right] \\ \mathrm{Now}, \\ \left(X+Y\right)-\left(X-Y\right)=\left[\begin{array}{cc}5& 2\\ 0& 9\end{array}\right]-\left[\begin{array}{cc}3& 6\\ 0& -1\end{array}\right] \\ \Rightarrow X+Y-X+Y=\left[\begin{array}{cc}5-3& 2-6\\ 0-0& 9+1\end{array}\right] \\ \Rightarrow 2Y=\left[\begin{array}{cc}2& -4\\ 0& 10\end{array}\right] \\ \Rightarrow Y=\frac{1}{2}\left[\begin{array}{cc}2& -4\\ 0& 10\end{array}\right] \\ \Rightarrow Y=\left[\begin{array}{cc}1& -2\\ 0& 5\end{array}\right] \\ \therefore X=\left[\begin{array}{cc}4& 4\\ 0& 4\end{array}\right]\mathrm{and}Y=\left[\begin{array}{cc}1& -2\\ 0& 5\end{array}\right] \end{gathered}$$