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Matrics — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsMatrics4 Marks
Question
Solve the following equations by inversion method $:x + y = 4, 2x – y = 5$
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Answer

$x + y = 4, 2x – y = 5$
The given equations can be written in the matrix form as $:\left[\begin{array}{cc}1 & 1 \\ 2 & -1\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}4 \\ 5\end{array}\right]$
This is of the form $AX = B$
$\Rightarrow X$
$\Rightarrow A^{-1}B$
$\begin{array}{l}A=\left[\begin{array}{cc}1 & 1 \\ 2 & -1\end{array}\right] \end{array} $
$ |A|=-1-2=-3 \neq 0$
$\begin{array}{l}\operatorname{Adj} A=\left[\begin{array}{cc}-1 & -1 \\ -2 & 1\end{array}\right] \end{array} $
$ =\frac{1}{-3}\left[\begin{array}{cc}-1 & -1 \\ -2 & 1\end{array}\right]$
$=\left[\begin{array}{cc}1 & 1 \\ 2 & -1\end{array}\right]  $
$ X=A^{-1} B=\left[\begin{array}{l}\mathrm{x} \\ \mathrm{y}\end{array}\right]=\left[\begin{array}{cc}1 & 1 \\ 2 & -1\end{array}\right]\left[\begin{array}{l}4 \\ 5\end{array}\right]  $
$ =\left[\begin{array}{cc}4 & 5 \\ 8 & -5\end{array}\right]$
$=\left[\begin{array}{l}9 \\ 3\end{array}\right]  $
$ =\left[\begin{array}{l}3 \\ 1\end{array}\right]$
By equality of matrices.
$x = 3, y = 1$

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