Question

Express the following equation and solve them by finding the inverse of coefficient matrix using adjoining method.
$4x+2y-z=3$
$x-2y+z=-8$
$2x-y+z=-7$

Answer 1

$\begin{array}{c}4x+2y-z=3\\ x-2y+z=-8\\ 2x-y+z=-7\\ \Rightarrow A=\left[\begin{array}{c}42-1\\ 1-21\\ 2-11\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}3\\ -8\\ -7\end{array}\right]\\ \Rightarrow \left|A\right|=4\left(-1\right)-2\left(-1\right)-1\left(3\right)\\ \Rightarrow \left|A\right|=-4+2-3=-5\ne 0\\ \Rightarrow Aisinvertible,\left(X=A^{-1}B\right)\\ Thencofactorof\left|A\right|are\\ \Rightarrow a_{11}=-1,a_{12}=1,a_{13}=3\\ \Rightarrow a_{21}=-1,a_{22}=6,a_{23}=8\\ \Rightarrow a_{31}=0,a_{32}=-5,a_{33}=-10\\ \Rightarrow adjA=\left[\begin{array}{c}-1-10\\ 16-5\\ 38-10\end{array}\right]\\ \Rightarrow X=A^{-1}B\\ \Rightarrow \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\frac{-1}{5}\left[\begin{array}{c}-1-10\\ 16-5\\ 3810\end{array}\right]\left[\begin{array}{c}3\\ -8\\ -7\end{array}\right]\\ \Rightarrow \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\frac{-1}{5}\left[\begin{array}{c}5\\ -10\\ 15\end{array}\right]=\left[\begin{array}{c}-1\\ 2\\ 3\end{array}\right]\\ \therefore x=-1,y=2andz=-3Ans.\end{array}$