Question

Solve the following system of linear equations using matrix method:
$3x+y+z=1,2x+2z=0,5x+y+2z=2$

Answer 1

Writing the given equation in the matrix form $AX=B$ as
$\left[\begin{array}{ccc}3& 1& 1\\ 2& 0& 2\\ 5& 1& 2\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ 0\\ 2\end{array}\right]$
where,

$A=\left[\begin{array}{ccc}3& 1& 1\\ 2& 0& 2\\ 5& 1& 2\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ and $B=\left[\begin{array}{c}1\\ 0\\ 2\end{array}\right]$

$\therefore$ $\left|A\right|=\left|\begin{array}{ccc}3& 1& 1\\ 2& 0& 2\\ 5& 1& 2\end{array}\right|=3(0-2)-1(4-10)+1(2-0)=2\ne 0$
$\therefore$ $A$ is non-singular
$\therefore$ The system has the unique solution $X=A^{-1}B$

$A_{11}=-2,A_{12}=6,A_{13}=2,A_{21}=-1,A_{22},A_{23}=2,A_{31}=2,A_{32}=-4,A_{33}=-2$

$\therefore adjA=\left[\begin{array}{ccc}{A}_{11}& A_{21}& A_{31}\\ A_{12}& A_{22}& A_{32}\\ A_{13}& A_{23}& A_{33}\end{array}\right]=\left[\begin{array}{ccc}-2& -1& 2\\ 6& 1& -4\\ 2& 2& -2\end{array}\right]$

$A^{-1}=\frac{adjA}{\left|A\right|}=\frac{1}{2}\left[\begin{array}{ccc}-2& -1& 2\\ 6& 1& -4\\ 2& 2& -2\end{array}\right]=\left[\begin{array}{ccc}-1& -1/2& 1\\ 3& 1/2& -2\\ 1& 1& -1\end{array}\right]$

Now,

$AX=B\Rightarrow X=A^{-1}B$

$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{ccc}-1& -1/2& 1\\ 3& 1/2& -2\\ 1& 1& -1\end{array}\right]\left[\begin{array}{c}1\\ 0\\ 2\end{array}\right]=\left[\begin{array}{c}1\\ -1\\ -1\end{array}\right]$
$x=1,y=-1,z=-1$