Question

Evaluate the following system of equation by Gaussian Elimination method

x1+5x2+2x3=9

x1+x2+7x3=6

-3x2+4x3=-2

Answer 1

Augmented matrix

$$A=\begin{pmatrix} 1 & 5 & 2 & \ \ 9\\ 1 & 1 & 7 & \ \ 6\\ 0 & -3 & 4 & \ \ -2\\ \end{pmatrix}$$

$R_2=R_2-R_1$

$$\begin{pmatrix} 1 & 5 & 2 & \ \ 9\\ 0 & -4 & 5 & \ \ -3\\ 0 & -3 & 4 & \ \ -2\\ \end{pmatrix}$$

$R_2=-\dfrac{R_2}{4}$

$$\begin{pmatrix} 1 & 5 & 2 & \ \ 9\\ 0 & 1 & -5/4 & \ \ 3/4\\ 0 & -3 & 4 & \ \ -2\\ \end{pmatrix}$$

$R_1=R_1-5R_2$

$$\begin{pmatrix} 1 & 0 & 33/4 & \ \ 21/4\\ 0 & 1 & -5/4 & \ \ 3/4\\ 0 & -3 & 4 & \ \ -2\\ \end{pmatrix}$$

$R_3=R_3+3R_2$

$$\begin{pmatrix} 1 & 0 & 33/4 & \ \ 21/4\\ 0 & 1 & -5/4 & \ \ 3/4\\ 0 & 0 & 1/4 & \ \ 1/4\\ \end{pmatrix}$$

$R_3=4R_3$

$$\begin{pmatrix} 1 & 0 & 33/4 & \ \ 21/4\\ 0 & 1 & -5/4 & \ \ 3/4\\ 0 & 0 & 1 & \ \ 1\\ \end{pmatrix}$$

$R_1=R_1-\dfrac{33R_3}{4}$

$$\begin{pmatrix} 1 & 0 & 0 & \ \ -3\\ 0 & 1 & -5/4 & \ \ 3/4\\ 0 & 0 & 1 & \ \ 1\\ \end{pmatrix}$$

$R_2=R_2+\dfrac{5R_3}{4}$

$$\begin{pmatrix} 1 & 0 & 0 & \ \ -3\\ 0 & 1 & 0 & \ \ 2\\ 0 & 0 & 1 & \ \ 1\\ \end{pmatrix}$$

$$x_1=-3, x_2=2, x_3=1$$

$(-3,2,1)$