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Question
Find matrices X and Y, if 2X − Y = `[[6 -6 0],[-4 2 1]]`and X + 2Y =`[[3 2 5],[-2 1 -7 ]]`
Solution
\[Given: \hspace{0.167em} \left( 2X - Y \right) = \begin{bmatrix}6 & - 6 & 0 \\ - 4 & 2 & 1\end{bmatrix} . . . \left( 1 \right)\]
\[ \left( X + 2Y \right) = \begin{bmatrix}3 & 2 & 5 \\ - 2 & 1 & - 7\end{bmatrix} . . . \left( 2 \right)\]
Multiplying eq. (1) by eq. (2), we get
\[2\left( 2X - Y \right) = 2\begin{bmatrix}6 & - 6 & 0 \\ - 4 & 2 & 1\end{bmatrix}\]
\[ \Rightarrow 4X - 2Y = \begin{bmatrix}12 & - 12 & 0 \\ - 8 & 4 & 2\end{bmatrix} . . . \left( 3 \right)\]
From eq. (3) and eq. (4) , we get
\[ \left( 4X - 2Y \right) + \left( X + 2Y \right) = \begin{bmatrix}12 & - 12 & 0 \\ - 8 & 4 & 2\end{bmatrix} + \begin{bmatrix}3 & 2 & 5 \\ - 2 & 1 & - 7\end{bmatrix}\]
\[ \Rightarrow 5X = \begin{bmatrix}12 + 3 & - 12 + 2 & 0 + 5 \\ - 8 - 2 & 4 + 1 & 2 - 7\end{bmatrix}\]
\[ \Rightarrow 5X = \begin{bmatrix}15 & - 10 & 5 \\ - 10 & 5 & - 5\end{bmatrix}\]
\[ \Rightarrow X = \frac{1}{5}\begin{bmatrix}15 & - 10 & 5 \\ - 10 & 5 & - 5\end{bmatrix}\]
\[ \Rightarrow X = \begin{bmatrix}3 & - 2 & 1 \\ - 2 & 1 & - 1\end{bmatrix}\]
Putting the value of X in eq . ( 2 ), we get
\[\left( X + 2Y \right) = \begin{bmatrix}3 & 2 & 5 \\ - 2 & 1 & - 7\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}3 & - 2 & 1 \\ - 2 & 1 & - 1\end{bmatrix} + 2Y = \begin{bmatrix}3 & 2 & 5 \\ - 2 & 1 & - 7\end{bmatrix}\]
\[ \Rightarrow 2Y = \begin{bmatrix}3 & 2 & 5 \\ - 2 & 1 & - 7\end{bmatrix} - \begin{bmatrix}3 & - 2 & 1 \\ - 2 & 1 & - 1\end{bmatrix}\]
\[ \Rightarrow 2Y = \begin{bmatrix}3 - 3 & 2 + 2 & 5 - 1 \\ - 2 + 2 & 1 - 1 & - 7 + 1\end{bmatrix}\]
\[ \Rightarrow Y = \begin{bmatrix}0 & 2 & 2 \\ 0 & 0 & - 3\end{bmatrix}\]
\[\]
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