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प्रश्न
find x, y , a and b if \[\begin{bmatrix}3x + 4y & 2 & x - 2y \\ a + b & 2a - b & - 1\end{bmatrix} = \begin{bmatrix}2 & 2 & 4 \\ 5 & - 5 & - 1\end{bmatrix}\]
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उत्तर
since the corresponding elements of two equal matrices are equal,
\[\begin{bmatrix}3x + 4y & 2 & x - 2y \\ a + b & 2a - b & - 1\end{bmatrix} = \begin{bmatrix}2 & 2 & 4 \\ 5 & - 5 & - 1\end{bmatrix}\]
\[\]
\[ \Rightarrow 3x + 4y = 2 . . . \left( 1 \right) \]
\[ \Rightarrow x - 2y = 4\]
⇒ x = 4 + 2y . . . ( 2)
\[\]`
`\text[Putting the value of x in eq . 1 , we get `
\[3\left( 4 + 2y \right) + 4y = 2 \]
\[ \Rightarrow 12 + 6y + 4y = 2 \]
\[ \Rightarrow 12 + 10y = 2 \]
\[ \Rightarrow 10y = 2 - 12\]
\[ \Rightarrow 10y = - 10\]
\[\]
\[ \Rightarrow y = \frac{- 10}{10} = - 1\]
\[\]
`\text[Putting the value of y in eq .2 ,we get`
\[x = 4 + 2\left( - 1 \right) \]
\[ \Rightarrow x = 4 - 2 = 2\]
\[\]
\[a + b = 5 \]
\[ \Rightarrow a = 5 - b . . . \left( 3 \right)\]
\[ \Rightarrow 2a - b = - 5 . . . \left( 4 \right)\]
\[ \]
`\text[Putting the value a in eq . 4, we get`
\[2\left( 5 - b \right) - b = - 5\]
\[ \Rightarrow 10 - 2b - b = - 5\]
\[ \Rightarrow 10 - 3b = - 5\]
\[ \Rightarrow - 3b = - 15 \]
\[ \Rightarrow b = \frac{- 15}{- 3} \]
\[ \Rightarrow b = 5\]
\[\]
`\text[Putting the value of b in eq . 3, we get`
\[ a = 5 - 5\]
\[ \Rightarrow a = 0\]
∴ x= 2 , y= 1, a = 0 and b = 5
