Finding Matrices X and Y

Solution:

Step 1: Eliminate \(Y\)

Multiply second equation by 1 and first by 2: \[ 4X – 2Y = \begin{bmatrix}12 & -12 & 0 \\ -8 & 4 & 2\end{bmatrix} \] \[ X + 2Y = \begin{bmatrix}3 & 2 & 5 \\ -2 & 1 & -7\end{bmatrix} \] Add both: \[ 5X = \begin{bmatrix} 12+3 & -12+2 & 0+5 \\ -8+(-2) & 4+1 & 2+(-7) \end{bmatrix} = \begin{bmatrix} 15 & -10 & 5 \\ -10 & 5 & -5 \end{bmatrix} \] \[ X = \begin{bmatrix} 3 & -2 & 1 \\ -2 & 1 & -1 \end{bmatrix} \]

Step 2: Find \(Y\)

Substitute into \(X + 2Y = B\): \[ 2Y = \begin{bmatrix}3 & 2 & 5 \\ -2 & 1 & -7\end{bmatrix} – \begin{bmatrix}3 & -2 & 1 \\ -2 & 1 & -1\end{bmatrix} = \begin{bmatrix} 0 & 4 & 4 \\ 0 & 0 & -6 \end{bmatrix} \] \[ Y = \begin{bmatrix} 0 & 2 & 2 \\ 0 & 0 & -3 \end{bmatrix} \]

Final Answer:

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