Question

Express the following matrix equation & solve by finding the inverse of coefficient matrix by elementry transformation method.
$x+y+z=6,x-y+2z=5,2x+y-z=1$

Answer 1

$A=\left[\begin{array}{ccc}1& 1& 1\\ 1& -1& 2\\ 2& 1& -1\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}6\\ 5\\ 1\end{array}\right]AX=BX=A^{-1}BAdjA=\left[\begin{array}{ccc}-1& 2& 3\\ 5& -3& -1\\ 3& 1& -2\end{array}\right]\left|A\right|=1(1-2)-1(-1-4)+1(1+2)=-1+5+3=7A^{-1}=\frac{AdjA}{\left|A\right|}=\frac{1}{7}\left[\begin{array}{ccc}-1& 2& 3\\ 5& -3& -1\\ 3& 1& -2\end{array}\right]\therefore X=\frac{1}{7}\left[\begin{array}{ccc}-1& 2& 3\\ 5& -3& -1\\ 3& 1& -2\end{array}\right]\left[\begin{array}{c}6\\ 5\\ 1\end{array}\right]=\frac{1}{7}\left[\begin{array}{c}-6+10+3\\ 30-15-1\\ 18+5-2\end{array}\right]=\frac{1}{7}\left[\begin{array}{c}7\\ 14\\ 21\end{array}\right]=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]\therefore x=1,y=2,z=3$