Question

A homogeneous system has a non-trivial solution, then ..
(a) All rows contain pivot position.
(b) The system has a free variables.
(c) The solutions must be unique.
(d) None.​

Answer 1

..Homogeneous Nonhomogeneous. Homogeneous System: Nontrivial Solutions. The homogeneous system Ax = 0 always has the trivial solution, x = 0. Nontrivial Solution Nonzero vector solutions are called nontrivial solutions.

Answer 2

A homogeneous system has a non-trivial solution, then (b) The system has a free variable.

• A homogeneous system refers to a system of linear equations in which all constants on the R.H.S (right-hand side) of the equation equals zero.
• The solution of a homogeneous system of equations is always x = 0. This solution is also called the trivial solution.
• Any solution that is not equal to zero is called a nontrivial solution.
• For a homogeneous system to have a non-trivial solution the equation must have at least one free variable or it should not have any pivot position. This is an absolute condition for a  homogeneous system to have a non-trivial solution.