Question

Solve the following system of equations by matrix method:
 x + y + z = 6
x + 2z = 7
3x + y + z = 12

Answer 1

Here, 
$$A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 2\\ 3& 1& 1\end{array}\right]$$
$$\left|A\right|=\left|\begin{array}{ccc}1& 1& 1\\ 1& 0& 2\\ 3& 1& 1\end{array}\right|$$
$$=1(0-2)-1(1-6)+1(1-0)$$
$$=-2+5+1$$
$$=4$$
$${\text{ Let }C}_{ij}{\text{ be the cofactors of the elements a }}_{ij}\text{ in }A\left[a_{ij}\right].\text{ Then,}$$
$$C_{11}={(-1)}^{1+1}\left|\begin{array}{cc}0& 2\\ 1& 1\end{array}\right|=-2,C_{12}={(-1)}^{1+2}\left|\begin{array}{cc}1& 2\\ 3& 1\end{array}\right|=5,C_{13}={(-1)}^{1+3}\left|\begin{array}{cc}1& 0\\ 3& 1\end{array}\right|=1$$
$$C_{21}={(-1)}^{2+1}\left|\begin{array}{cc}1& 1\\ 1& 1\end{array}\right|=0,C_{22}={(-1)}^{2+2}\left|\begin{array}{cc}1& 1\\ 3& 1\end{array}\right|=-2,C_{23}={(-1)}^{2+3}\left|\begin{array}{cc}1& 1\\ 3& 1\end{array}\right|=2$$
$$C_{31}={(-1)}^{3+1}\left|\begin{array}{cc}1& 1\\ 0& 2\end{array}\right|=2,C_{32}={(-1)}^{3+2}\left|\begin{array}{cc}1& 1\\ 1& 2\end{array}\right|=-1,C_{33}={(-1)}^{3+3}\left|\begin{array}{cc}1& 1\\ 1& 0\end{array}\right|=-1$$
$$adjA={\left[\begin{array}{ccc}-2& 5& 1\\ 0& -2& 2\\ 2& -1& -1\end{array}\right]}^T$$
$$=\left[\begin{array}{ccc}-2& 0& 2\\ 5& -2& -1\\ 1& 2& -1\end{array}\right]$$
$$\Rightarrow A^{-1}=\frac{1}{\left|A\right|}adjA$$
$$=\frac{1}{4}\left[\begin{array}{ccc}-2& 0& 2\\ 5& -2& -1\\ 1& 2& -1\end{array}\right]$$
$$X=A^{-1}B$$
$$\Rightarrow \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\frac{1}{4}\left[\begin{array}{ccc}-2& 0& 2\\ 5& -2& -1\\ 1& 2& -1\end{array}\right]\left[\begin{array}{c}6\\ 7\\ 12\end{array}\right]$$
$$\Rightarrow \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\frac{1}{4}\left[\begin{array}{c}-12+0+24\\ 30-14-12\\ 6-14-12\end{array}\right]$$
$$\Rightarrow \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\frac{1}{4}\left[\begin{array}{c}12\\ 4\\ -20\end{array}\right]$$
$$\Rightarrow x=\frac{12}{4},y=\frac{4}{4}\text{ and }z=\frac{-20}{4}$$
$$\therefore x=3,y=1\text{ and }z=-5$$