A homogeneous system is of the form
To have a solution, .
To have infinitely many solutions, at least one of the equations is a scalar multiple of another.
give an example of a 3x3 homogeneous system of linear equations AX=0 which has infinitely many solutions.
a system like this?
my teacher told me this is a cheesy answer to give if it were to appear on a final exam, but is correct.
how so? what do they mean exactly by a homogeneous system with infinitely many solutions?
Yes, that is true.
I notice that Prove It said: "To have a solution, .
To have infinitely many solutions, at least one of the equations is a scalar multiple of another. "
To have a unique solution the determinant must not be 0. Since a homogeneous system of equations always has the trivial solution (all unknowns equal to 0), a homogenous system of equations has an infinite number of solutions if and only if its determinant is equal to 0.