Question

Find matrices X and Y, if 2X − Y = $\left[\begin{array}{ccc}6& -6& 0\\ -4& 2& 1\end{array}\right]$ and X + 2Y = $\left[\begin{array}{ccc}3& 2& 5\\ -2& 1& -7\end{array}\right]$

Answer 1

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}\left(2X-Y\right)=\left[\begin{array}{ccc}6& -6& 0\\ -4& 2& 1\end{array}\right]...\left(1\right) \\ \left(X+2Y\right)=\left[\begin{array}{ccc}3& 2& 5\\ -2& 1& -7\end{array}\right]...\left(2\right) \\ \mathrm{Multiplying}\mathrm{eq}.\left(1\right)\mathrm{by}\mathrm{eq}.\left(2\right),\mathrm{we}\mathrm{get} \\ 2\left(2X-Y\right)=2\left[\begin{array}{ccc}6& -6& 0\\ -4& 2& 1\end{array}\right] \\ \Rightarrow 4X-2Y=\left[\begin{array}{ccc}12& -12& 0\\ -8& 4& 2\end{array}\right]...\left(3\right) \\ \mathrm{From}\mathrm{eq}.\left(3\right)\mathrm{and}\mathrm{eq}.\left(4\right),\mathrm{we}\mathrm{get} \\ \left(4X-2Y\right)+\left(X+2Y\right)=\left[\begin{array}{ccc}12& -12& 0\\ -8& 4& 2\end{array}\right]+\left[\begin{array}{ccc}3& 2& 5\\ -2& 1& -7\end{array}\right] \\ \Rightarrow 5X=\left[\begin{array}{ccc}12+3& -12+2& 0+5\\ -8-2& 4+1& 2-7\end{array}\right] \\ \Rightarrow 5X=\left[\begin{array}{ccc}15& -10& 5\\ -10& 5& -5\end{array}\right] \\ \Rightarrow X=\frac{1}{5}\left[\begin{array}{ccc}15& -10& 5\\ -10& 5& -5\end{array}\right] \\ \Rightarrow X=\left[\begin{array}{ccc}3& -2& 1\\ -2& 1& -1\end{array}\right] \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}X\mathrm{in}\mathrm{eq}.\left(2\right),\mathrm{we}\mathrm{get} \\ \left(X+2Y\right)=\left[\begin{array}{ccc}3& 2& 5\\ -2& 1& -7\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}3& -2& 1\\ -2& 1& -1\end{array}\right]+2Y=\left[\begin{array}{ccc}3& 2& 5\\ -2& 1& -7\end{array}\right] \\ \Rightarrow 2Y=\left[\begin{array}{ccc}3& 2& 5\\ -2& 1& -7\end{array}\right]-\left[\begin{array}{ccc}3& -2& 1\\ -2& 1& -1\end{array}\right] \\ \Rightarrow 2Y=\left[\begin{array}{ccc}3-3& 2+2& 5-1\\ -2+2& 1-1& -7+1\end{array}\right] \\ \Rightarrow Y=\left[\begin{array}{ccc}0& 2& 2\\ 0& 0& -3\end{array}\right] \end{gathered}$$