Question

Use matrix method to solve the following the equations
$x+y+z=3,$

$2x-y+z=2,$

$x-2y+3z=2$

Find the value of $x+y-2z?$

Answer 1

$AX=B$

$\left[\begin{array}{ccc}1& 1& 1\\ 2& -1& 1\\ 1& -2& 3\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}3\\ 2\\ 2\end{array}\right]$

$|A|=\left|\begin{array}{ccc}1& 1& 1\\ 2& -1& 1\\ 1& -2& 3\end{array}\right|=-9$

$adjA=\left[\begin{array}{ccc}-1& -5& 2\\ -5& 2& 1\\ -3& 3& -3\end{array}\right]$

$A^{-1}=\frac{-1}{9}\left[\begin{array}{ccc}-1& -5& 2\\ -5& 2& 1\\ -3& 3& -3\end{array}\right]$

$X=A^{-1}B=\frac{-1}{9}\left[\begin{array}{ccc}-1& -5& 2\\ -5& 2& 1\\ -3& 3& -3\end{array}\right]\left[\begin{array}{c}3\\ 2\\ 2\end{array}\right]=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$

$\Rightarrow x=1,y=1$ and $z=1$

$\therefore x+y-2z=0$