Solving for X and Y

Solution:

Step 1: Eliminate Y

Multiply first equation by 2 and second by 3: \[ 4X + 6Y = \begin{bmatrix}4 & 6 \\ 8 & 0\end{bmatrix} \] \[ 9X + 6Y = \begin{bmatrix}-6 & 6 \\ 3 & -15\end{bmatrix} \] Subtract: \[ 5X = \begin{bmatrix} -6-4 & 6-6 \\ 3-8 & -15-0 \end{bmatrix} = \begin{bmatrix} -10 & 0 \\ -5 & -15 \end{bmatrix} \] \[ X = \begin{bmatrix} -2 & 0 \\ -1 & -3 \end{bmatrix} \]

Step 2: Find Y

Substitute into \(2X + 3Y = A\): \[ 2X = \begin{bmatrix} -4 & 0 \\ -2 & -6 \end{bmatrix} \] \[ 3Y = \begin{bmatrix} 2 & 3 \\ 4 & 0 \end{bmatrix} – \begin{bmatrix} -4 & 0 \\ -2 & -6 \end{bmatrix} = \begin{bmatrix} 6 & 3 \\ 6 & 6 \end{bmatrix} \] \[ Y = \begin{bmatrix} 2 & 1 \\ 2 & 2 \end{bmatrix} \]

Final Answer:

\[ X=\begin{bmatrix}-2 & 0 \\ -1 & -3\end{bmatrix}, \quad Y=\begin{bmatrix}2 & 1 \\ 2 & 2\end{bmatrix} \]