Question

Solve the following equations by matrix method.
$2x+y+z=1,x-2y-3z=1,3x+2y+4z=5$

Answer 1

The given system of equations can be written in matrix form as: $AX=b$ where

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

$b=\left[\begin{array}{c}1\\ 1\\ 5\end{array}\right]$ and

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

$|A|=-9$

So, we have,

$cof\left(A\right)=\left[\begin{array}{ccc}-2& -13& 8\\ -2& 5& -1\\ -1& 7& -5\end{array}\right]$

$Adj\left(A\right)=\left[cof\right(A){]}^T=\left[\begin{array}{ccc}-2& -2& -1\\ -13& 5& 7\\ 8& -1& -5\end{array}\right]$

$A^{-1}=\frac{1}{|A|}Adj\left(A\right)$

$\therefore A^{-1}=\frac{1}{-9}\left[\begin{array}{ccc}-2& -2& -1\\ -13& 5& 7\\ 8& -1& -5\end{array}\right]$

$X=A^{-1}b=\left[\begin{array}{c}1\\ -3\\ 2\end{array}\right]$

Thus, the solution is $x=1,y=-3,z=2$