Finding Matrices X and Y

Solution:

Step 1: Add the equations

\[ (X+Y)+(X-Y)=2X \] \[ = \begin{bmatrix}5 & 2 \\ 0 & 9\end{bmatrix} + \begin{bmatrix}3 & 6 \\ 0 & -1\end{bmatrix} = \begin{bmatrix}8 & 8 \\ 0 & 8\end{bmatrix} \] \[ \Rightarrow 2X= \begin{bmatrix}8 & 8 \\ 0 & 8\end{bmatrix} \] \[ X= \begin{bmatrix}4 & 4 \\ 0 & 4\end{bmatrix} \]

Step 2: Subtract the equations

\[ (X+Y)-(X-Y)=2Y \] \[ = \begin{bmatrix}5 & 2 \\ 0 & 9\end{bmatrix} – \begin{bmatrix}3 & 6 \\ 0 & -1\end{bmatrix} = \begin{bmatrix}2 & -4 \\ 0 & 10\end{bmatrix} \] \[ \Rightarrow 2Y= \begin{bmatrix}2 & -4 \\ 0 & 10\end{bmatrix} \] \[ Y= \begin{bmatrix}1 & -2 \\ 0 & 5\end{bmatrix} \]

Final Answer:

\[ X=\begin{bmatrix}4 & 4 \\ 0 & 4\end{bmatrix}, \quad Y=\begin{bmatrix}1 & -2 \\ 0 & 5\end{bmatrix} \]