Question

If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y.

$2X+3Y=\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right],3X+2Y=\left[\begin{array}{cc}-2& 2\\ 1& -5\end{array}\right]$

Answer 1

We have,
$$\begin{gathered} 3\left(2X+3Y\right)-2\left(3X+2Y\right)=3\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]-2\left[\begin{array}{cc}-2& 2\\ 1& -5\end{array}\right] \\ \Rightarrow 6X+9Y-6X-4Y=\left[\begin{array}{cc}6& 9\\ 12& 0\end{array}\right]+\left[\begin{array}{cc}4& -4\\ -2& 10\end{array}\right] \\ \Rightarrow 5Y=\left[\begin{array}{cc}6+4& 9-4\\ 12-2& 0+10\end{array}\right] \\ \Rightarrow Y=\frac{1}{5}\left[\begin{array}{cc}10& 5\\ 10& 10\end{array}\right] \\ \Rightarrow Y=\left[\begin{array}{cc}2& 1\\ 2& 2\end{array}\right]...\left(1\right) \end{gathered}$$


Also,
$$\begin{gathered} 2\left(2X+3Y\right)-3\left(3X+2Y\right)=2\left[\begin{array}{cc}2& 3\\ 4& 0\end{array}\right]-3\left[\begin{array}{cc}-2& 2\\ 1& -5\end{array}\right] \\ \Rightarrow 4X+6Y-9X-6Y=\left[\begin{array}{cc}4& 6\\ 8& 0\end{array}\right]+\left[\begin{array}{cc}6& -6\\ -3& 15\end{array}\right] \\ \Rightarrow -5X=\left[\begin{array}{cc}6+4& 6-6\\ 8-3& 0+15\end{array}\right] \\ \Rightarrow X=\frac{1}{-5}\left[\begin{array}{cc}10& 0\\ 5& 15\end{array}\right] \\ \Rightarrow X=\left[\begin{array}{cc}-2& 0\\ -1& -3\end{array}\right]...\left(2\right) \end{gathered}$$

From (1) and (2), we get

$X=\left[\begin{array}{cc}-2& 0\\ -1& -3\end{array}\right]\mathrm{and}Y=\left[\begin{array}{cc}2& 1\\ 2& 2\end{array}\right]$.