Question

Determine if the system has a nontrivial solution. Try to use as few row operations as possible.

4x₁-6x₂+13x₃=0
-4X₁-10X₂-X₃=0
8x₁+4x₂+14x₃ 0

Answer 1

To determine if the system has a nontrivial solution, we examine the reduced row echelon form of the coefficient matrix. The system is consistent and has a nontrivial solution x₁ = -2 and x₂ = -9.

Step-by-step explanation:

To determine if the system has a nontrivial solution, we need to examine the coefficient matrix and its reduced row echelon form. Let's set up the augmented matrix for the system:

| 4 -6 13 | 0 |
|-4 -10 -1 | 0 |
| 8 4 14 | 0 |

Performing elementary row operations, we can reduce the augmented matrix to:

| 1 0 2 | 0 |
| 0 1 9 | 0 |
| 0 0 0 | 0 |

The reduced row echelon form shows that the system is consistent and has a nontrivial solution. The nontrivial solution is x₁ = -2 and x₂ = -9. Therefore, the system has a nontrivial solution.