Question

Consider the following System of equations:

3𝑥 + 2𝑦 + 𝑧 = 3

2𝑥 + 𝑦 + 𝑧 = 0

6𝑥 + 2𝑦 + 4𝑧 = 6

a. Apply Gaussian elimination method to reduce the system to triangular form.

b. What do you observe from your answer in part (a) above?

Answer 1

a.

Augmented matrix is

$$\begin{pmatrix} 3 & 2 & 1 & & 3 \\ 2 & 1 & 1 & & 0 \\ 6 & 2 & 4 & & 6 \\ \end{pmatrix}$$

$R_1=R_1/3$

$$\begin{pmatrix} 1 & 2/3 & 1/3 & & 1 \\ 2 & 1 & 1 & & 0 \\ 6 & 2 & 4 & & 6 \\ \end{pmatrix}$$

$R_2=R_2-2R_1$

$$\begin{pmatrix} 1 & 2/3 & 1/3 & & 1 \\ 0& -1/3 & 1/3 & & -2\\ 6 & 2 & 4 & & 6 \\ \end{pmatrix}$$

$R_3=R_3-6R_1$

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

$R_2=-3R_2$

$$\begin{pmatrix} 1 & 2/3 & 1/3 & & 1 \\ 0 & 1 & -1 & & 6\\ 0 & -2 & 2 & & 0 \\ \end{pmatrix}$$

$R_1=R_1-2R_2/3$

$$\begin{pmatrix} 1 & 0 & 1 & & -3\\ 0 & 1 & -1 & & 6\\ 0 & -2 & 2 & & 0 \\ \end{pmatrix}$$

$R_3=R_3+2R_2$

$$\begin{pmatrix} 1 & 0 & 1 & & -3\\ 0 & 1 & -1 & & 6\\ 0 & 0 & 0 & & 12 \\ \end{pmatrix}$$

$R_3=R_3/12$

$$\begin{pmatrix} 1 & 0 & 1 & & -3\\ 0 & 1 & -1 & & 6\\ 0 & 0 & 0 & & 1 \\ \end{pmatrix}$$

$R_1=R_1+3R_3$

$$\begin{pmatrix} 1 & 0 & 1 & & 0\\ 0 & 1 & -1 & & 6\\ 0 & 0 & 0 & & 1 \\ \end{pmatrix}$$

$R_2=R_2-6R_3$

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

b. The system is inconsistent. The given system has no solution.