Express the following equation and solve them by finding the inverse of coefficient matrix using adjoining method.
$x + y + z = 9$
$2 x + 5 y + 7 z = 52$
$2 x + y - z = 0$

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Solution

$X = ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦$

$B = ⎡ ⎢ ⎣ \begin{matrix} 9520 \end{matrix} ⎤ ⎥ ⎦$

$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 2 & 5 & 7 2 & 1 & - 1 \end{matrix} ⎤ ⎥ ⎦$

$\Rightarrow A^{- 1} = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 3 & - \frac{1}{2} & - \frac{1}{2} - 4 & \frac{3}{4} & \frac{5}{4} 2 & - \frac{1}{4} & - \frac{3}{4} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦$

$X = A^{- 1} B$

$⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 3 & - \frac{1}{2} & - \frac{1}{2} - 4 & \frac{3}{4} & \frac{5}{4} 2 & - \frac{1}{4} & - \frac{3}{4} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 9520 \end{matrix} ⎤ ⎥ ⎦$

$⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 135 \end{matrix} ⎤ ⎥ ⎦$

$∴ x = 1, y = 3, z = 5$

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