Question

Solve the following set of simultaneous equations by gauss elimination methos.

x - 2y + z = 3 ......(1)

x + 3z = 11 ......(2)

-2y + z = 1 .......(3)

Answer 1

Gauss elimination method

x - 2y + 2 = 3

x + 0y + 3z = 11

0x - 2y + z = 1

$\left[\begin{array}{ccc}1& -2& 1\\ 1& 0& 3\\ 0& -2& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}3\\ 11\\ 1\end{array}\right]$ ......(i)
A X B

Augmented matrix [A : B]

[A : B] = $\left[\begin{array}{ccccc}1& -2& 1& |& 3\\ 1& 0& 3& |& 11\\ 0& -2& 1& |& 1\end{array}\right]$

$\stackrel{R_2\to R_2\to R_1}{\to }$ $\left[\begin{array}{ccccc}1& -2& 1& |& 3\\ 0& 2& 2& |& 8\\ 0& -2& 1& |& 1\end{array}\right]$

$\stackrel{R_3\to R_3\to R_2}{\to }$ $\left[\begin{array}{ccccc}1& -2& 1& |& 3\\ 0& 2& 2& |& 8\\ 0& 0& 3& |& 9\end{array}\right]$ .....(ii)

$\because$ (i) & (ii) are equivalent so

AX = B $\Rightarrow$ $\left[\begin{array}{ccc}1& -2& 1\\ 1& 0& 3\\ 0& -2& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}3\\ 11\\ 1\end{array}\right]-\left[\begin{array}{ccc}1& -2& 1\\ 0& 2& 2\\ 0& 0& 3\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}3\\ 8\\ 9\end{array}\right]$

$\Rightarrow$ $textrm\begin{array}{c}x-2y+z=3\\ 2y+2z=8\\ 3z=9\end{array}$ $\Rightarrow$ x = 2, y = 1, z = 3