Question

Determine by inspection whether the given system has nontrivial solutions only the trivial solutions

1. 2x₁+3x₂- x₃ = 0
x₁- x₂+2 x₃ = 0

2. x₁+2x₂- x₃+2x₄ = 0
2x₁+x₂-+x₃-x₄= 0
3x₁-x₂-2x₃+3x₄ = 0

Answer 1

To determine whether a given system has nontrivial solutions or only trivial solutions, we need to examine the coefficients of the variables in each equation. If all the variables have a coefficient of zero, then the system has trivial solutions. Otherwise, it may have nontrivial solutions.

Step-by-step explanation:

To determine whether a given system has nontrivial solutions or only trivial solutions, we need to examine the coefficients of the variables in each equation. If all the variables have a coefficient of zero, then the system has trivial solutions. Otherwise, it may have nontrivial solutions.

Let's analyze the given systems:

1. 2x₁+3x₂- x₃ = 0 x₁- x₂+2 x₃ = 0
2. x₁+2x₂- x₃+2x₄ = 0 2x₁+x₂-+x₃-x₄= 0 3x₁-x₂-2x₃+3x₄ = 0

1. In the first system, the variables have non-zero coefficients, so it has nontrivial solutions.
2. In the second system, the variables also have non-zero coefficients, so it also has nontrivial solutions.