Write the solution set of the given homogeneous system in pa

Write the solution set of the given homogeneous system in parametric vector form. 2x_1 + 2x_2 + 4x_3 = 0 -4x_1 - 4x_2 - 8x_3 = 0 -3x_2 - 9x_3 = 0 where the solution set is x = [x_1 x_2 x_3] Select the correct choice below and fill in the answer box(es) within your choice. X = x_2[] + x_3[] x = [] x = x_3[] x = x_1[] + x_2[] + x_3[]

Solution

A, the coefficient matrix of the given linear system is

2

2

4

-4

-4

-8

0

-3

-9

We will reduce A to its RREF as under:

Multiply the 1st row by ½

Add 4 times the 1st row to the 2nd row

Interchange the 2nd row and the 3rd row

Multiply the 2nd row by -1/3

Add -1 times the 2nd row to the 1st row

Then the RREF of A is

1

0

-1

0

1

3

0

0

0

Thus, the given linear system is equivalent to:

x₁ –x₃ = 0 and x₂ +3x₃ = 0 so that x₁ =x₃ and x₂ = -3x₃ . Hence X = (x₁,x₂,x₃)^T = (x₃, -3x₃,x₃ )^T = x₃( 1,-3,1)^T Thus, option c is the correct answer.  

2	2	4
-4	-4	-8
0	-3	-9