Question

Determine if the system has a nontrivial solution. Try to use as few row operations as possible 4x1-6x2 + 13x3 = 0 -4x1-2x2-7x3 = 0 8x1 4x2+14x30 Choose the correct answer below. O A. O B. O C. The system has a nontrivial solution. It is impossible to determine. The system has only a trivial solution.

Answer 1

system is

augmented matrix is

4	-6	13
-4	-2	-7
8	4	14

convert into Reduced Row Eschelon Form...

Divide row1 by 4

1	-3/2	13/4
-4	-2	-7
8	4	14

Add (4 * row1) to row2

1	-3/2	13/4
0	-8	6
8	4	14

Add (-8 * row1) to row3

1	-3/2	13/4
0	-8	6
0	16	-12

Divide row2 by -8

1	-3/2	13/4
0	1	-3/4
0	16	-12

Add (-16 * row2) to row3

1	-3/2	13/4
0	1	-3/4
0	0	0

Add (3/2 * row2) to row1

1	0	17/8
0	1	-3/4
0	0	0

reduced system is

$\begin{pmatrix}1&0&\frac{17}{8}\\ 0&1&-\frac{3}{4}\\ 0&0&0\end{pmatrix}\begin{pmatrix}x_1\\ x_2\\ x_3\end{pmatrix}=0$

$x_1+\frac{17}{8}x_3=0.........{\color{Red} x_1=-\frac{17}{8}x_3}$

$x_2-\frac{3}{4}x_3=0.........{\color{Red} x_2=\frac{3}{4}x_3}$

${\color{Red} x_3=x_3}$ .................free variable

non trivial solution is

${\color{Red} \begin{pmatrix}x_1\\ x_2\\ x_3\end{pmatrix}=\begin{pmatrix}-\frac{17}{8}\\ \frac{3}{4}\\ 1\end{pmatrix}x_3}$