[Math] What does “multiple non-trivial solutions exists mean?”

homogeneous equationlinear algebrasystems of equations

I came across this question of solving system of linear non-homogeneous equations :

$$x+2y+z+4w=2$$
$$3x+6y+3z+12w=6$$

Options are :

  1. Only the trivial solution $x=y=z=w=0$ exists
  2. There is no solution
  3. A unique non-solution exists
  4. Multiple non-trivial solutions exists.

The answer is Multiple non-trivial solutions exists.

Can somebody explain me why and what does it mean? I always thought the rows in the matrix must be equal to the number of variables?

Best Answer

"Multiple non-trivial solutions exist": a solution is called nontrivial if it is not identically zero (like in your option 1). So this statement means there are at least two different solutions to that equation which are not that particular zero solution.

Edit (actually the trivial solution does not satisfy the equation(s), so it is not a solution).

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