Question

If the matrices \( X + Y = \begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix} \) and \( X - Y = \begin{bmatrix} 3 & 6 \\ 0 & -1 \end{bmatrix} \), then find the matrices \( X \) and \( Y \).

Hint:

To solve for matrices \( X \) and \( Y \), use matrix addition and subtraction, then divide by \( 2 \).

Answer 1

Adding and subtracting the given equations: \[ X + Y = \begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix}, \quad X - Y = \begin{bmatrix} 3 & 6 \\ 0 & -1 \end{bmatrix}. \] \[ 2X = \begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix} + \begin{bmatrix} 3 & 6 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 8 & 8 \\ 0 & 8 \end{bmatrix}. \] \[ X = \begin{bmatrix} 4 & 4 \\ 0 & 4 \end{bmatrix}. \] \[ 2Y = \begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix} - \begin{bmatrix} 3 & 6 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 2 & -4 \\ 0 & 10 \end{bmatrix}. \] \[ Y = \begin{bmatrix} 1 & -2 \\ 0 & 5 \end{bmatrix}. \]