# Solve the following system of equations by matrix method :

Given : system of linear equations

Given system of linear equations can be written in matrix form as

$\mathrm{AX}=\mathrm{B}\\$

Where

$A = \left[\begin{array}{ccc} 1 & -1 & 2 \\ 2 & -1 & 3 \\ 3 & 2 & -1 \end{array}\right]$$X= \left[\begin{array}{l} x \\ y \\ z \end{array}\right]$$B=\left[\begin{array}{c} 7 \\ 12 \\ 5 \end{array}\right]\\$

$|\mathrm{A}|=-2 \neq 0\Rightarrow A is invertible$

$Adj A =\left[\begin{array}{ccc} -5 & 11 & 7 \\ 3 & -7 & -5 \\ -1 & 1 & 1 \end{array}\right]^T\\$

$A^{-1}=\frac{Adj A}{|A|}$

$A^{-1}=\frac{1}{-2}\left[\begin{array}{ccc} -5 & 3 & -1 \\ 11 & -7 & 1 \\ 7 & -5 & 1 \end{array}\right]\\$

$X=A^{-1} B$

$X=\frac{1}{-2}\left[\begin{array}{ccc} -5 & 3 & -1 \\ 11 & -7 & 1 \\ 7 & -5 & 1 \end{array}\right]\left[\begin{array}{c} 7 \\ 12 \\ 5 \end{array}\right]$

$X=\left[\begin{array}{l} 2 \\ 1 \\ 3 \end{array}\right]\\$

$x=2, y=1, z=3$

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