Finding Matrices X and Y (3×3)
Question:
If \[ X – Y = \begin{bmatrix} 1 & 1 & 1\\ 1 & 1 & 0\\ 1 & 0 & 0 \end{bmatrix}, \quad X + Y = \begin{bmatrix} 3 & 5 & 1\\ -1 & 1 & 4\\ 11 & 8 & 0 \end{bmatrix} \] find matrices \(X\) and \(Y\).
If \[ X – Y = \begin{bmatrix} 1 & 1 & 1\\ 1 & 1 & 0\\ 1 & 0 & 0 \end{bmatrix}, \quad X + Y = \begin{bmatrix} 3 & 5 & 1\\ -1 & 1 & 4\\ 11 & 8 & 0 \end{bmatrix} \] find matrices \(X\) and \(Y\).
Solution:
Step 1: Add the equations
\[ (X+Y)+(X-Y)=2X \] \[ 2X = \begin{bmatrix} 3+1 & 5+1 & 1+1\\ -1+1 & 1+1 & 4+0\\ 11+1 & 8+0 & 0+0 \end{bmatrix} = \begin{bmatrix} 4 & 6 & 2\\ 0 & 2 & 4\\ 12 & 8 & 0 \end{bmatrix} \] \[ X = \begin{bmatrix} 2 & 3 & 1\\ 0 & 1 & 2\\ 6 & 4 & 0 \end{bmatrix} \]Step 2: Subtract the equations
\[ (X+Y)-(X-Y)=2Y \] \[ 2Y = \begin{bmatrix} 3-1 & 5-1 & 1-1\\ -1-1 & 1-1 & 4-0\\ 11-1 & 8-0 & 0-0 \end{bmatrix} = \begin{bmatrix} 2 & 4 & 0\\ -2 & 0 & 4\\ 10 & 8 & 0 \end{bmatrix} \] \[ Y = \begin{bmatrix} 1 & 2 & 0\\ -1 & 0 & 2\\ 5 & 4 & 0 \end{bmatrix} \]Final Answer:
\[ X= \begin{bmatrix} 2 & 3 & 1\\ 0 & 1 & 2\\ 6 & 4 & 0 \end{bmatrix}, \quad Y= \begin{bmatrix} 1 & 2 & 0\\ -1 & 0 & 2\\ 5 & 4 & 0 \end{bmatrix} \]