Question

If X − Y = $\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 0\\ 1& 0& 0\end{array}\right]$ and X + Y = $\left[\begin{array}{ccc}3& 5& 1\\ -1& 1& 4\\ 11& 8& 0\end{array}\right]$, find X and Y.

Answer 1

$$\begin{gathered} \mathrm{Here}, \\ X-Y+X+Y=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 0\\ 1& 0& 0\end{array}\right]+\left[\begin{array}{ccc}3& 5& 1\\ -1& 1& 4\\ 11& 8& 0\end{array}\right] \\ \Rightarrow 2X=\left[\begin{array}{ccc}1+3& 1+5& 1+1\\ 1-1& 1+1& 0+4\\ 1+11& 0+8& 0+0\end{array}\right] \\ \Rightarrow 2X=\left[\begin{array}{ccc}4& 6& 2\\ 0& 2& 4\\ 12& 8& 0\end{array}\right] \\ \Rightarrow X=\frac{1}{2}\left[\begin{array}{ccc}4& 6& 2\\ 0& 2& 4\\ 12& 8& 0\end{array}\right] \\ \Rightarrow X=\left[\begin{array}{ccc}2& 3& 1\\ 0& 1& 2\\ 6& 4& 0\end{array}\right] \\ \mathrm{Now}, \\ \left(X-Y\right)-\left(X+Y\right)=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 0\\ 1& 0& 0\end{array}\right]-\left[\begin{array}{ccc}3& 5& 1\\ -1& 1& 4\\ 11& 8& 0\end{array}\right] \\ \Rightarrow X-Y-X-Y=\left[\begin{array}{ccc}1-3& 1-5& 1-1\\ 1+1& 1-1& 0-4\\ 1-11& 0-8& 0-0\end{array}\right] \\ \Rightarrow -2Y=\left[\begin{array}{ccc}-2& -4& 0\\ 2& 0& -4\\ -10& -8& 0\end{array}\right] \\ \Rightarrow Y=-\frac{1}{2}\left[\begin{array}{ccc}-2& -4& 0\\ 2& 0& -4\\ -10& -8& 0\end{array}\right] \\ \Rightarrow Y=\left[\begin{array}{ccc}1& 2& 0\\ -1& 0& 2\\ 5& 4& 0\end{array}\right] \\ \therefore X=\left[\begin{array}{ccc}2& 3& 1\\ 0& 1& 2\\ 6& 4& 0\end{array}\right]\mathrm{and}Y=\left[\begin{array}{ccc}1& 2& 0\\ -1& 0& 2\\ 5& 4& 0\end{array}\right] \end{gathered}$$