Question

Solve the following system of equations by using Matrix inversion method.
$2x-y+3z=9,x+y+z=6,x-y+z=2$

Answer 1

Given

$2x-y+3z=9$ ………..$\left(1\right)$

$x+y+z=6$ …………..$\left(2\right)$

$x-y+z=2$ …………….$\left(3\right)$

using matrix inversion method

$AX=B$

$\left[\begin{array}{ccc}2& -1& 3\\ 1& 1& 1\\ 1& -1& 1\end{array}\right]$ $\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}9\\ 6\\ 2\end{array}\right]$

and $X=A^{-1}B$ where $A^{-1}=\frac{adj\left(A\right)}{|A|}$

Now $adjA=\left[\begin{array}{ccc}2& -2& -4\\ 0& -1& 1\\ -2& -1& 3\end{array}\right]$

$|A|=2(1+1)-1(1-1)+3(-1-1)$

$|A|=4-6$

$|A|=-2$.

Now $A^{-1}=\frac{-1}{2}\left[\begin{array}{ccc}2& -2& -4\\ 0& -1& 1\\ -2& -1& 3\end{array}\right]$

Now $X=A^{-1}B$

$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\frac{-1}{2}\left[\begin{array}{ccc}2& -2& -4\\ 0& -1& 1\\ -2& -1& 3\end{array}\right]\left[\begin{array}{c}9\\ 6\\ 2\end{array}\right]$

$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\frac{-1}{2}\left[\begin{array}{c}18-12-8\\ 0-6+2\\ -18-6+6\end{array}\right]=\frac{-1}{2}\left[\begin{array}{c}-2\\ -4\\ -18\end{array}\right]$

$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ 2\\ 9\end{array}\right]$

Hence $[x=1,y=2$ & $z=9]$.