homogeneous only trivial or infinite

Question

This question has three parts. There are similar questions on stack exchange, but if you read all of the questions, then you'll see that this is not a duplicate [at least not one that I could find.]

From the Oregon math website:

"A $n\times n$ homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions."

1. Does this imply that a homogeneous system $Ax=0$ has only a trivial solution or infinite number of solutions?

2. If not, then what values of the determinant for $A$ imply that there is a non-trivial, unique solution, for a homogeneous equation?

3. If it does imply that $Ax=0$ only has a trivial or infinite (free parameter) solution, isn't that kind of weird?

Answer 1

From the given theorem we have that

1. Does this imply that a homogeneous system $Ax=0$ has only a trivial solution or infinite number of solutions?

• recall that $\det A=0$ or $\det A \neq 0$

2. If not, then what values of the determinant for A imply that there is a non-trivial, unique solution, for a homogeneous equation?

• what about $\det A \neq 0$?

3. If it does imply that $Ax=0$ only has a trivial or infinite (free parameter) solution, isn't that kind of weird?

• it is not weird, it is a theorem

Answer 2

It is good to notice that a homogeneous system $Ax=0$ always has at least zero solution $x=0$. (So for homogeneous system you cannot have no solutions at all.)

If you have any solution $x$, than any scalar multiple $cx$ is also a solution of the same homogeneous system. So once you have at least one non-zero solution, you immediately get many other solutions. (If you are working with real numbers or some other infinite field, you get infinitely many solutions.)

Things are different if you look at systems which are not homogeneous - then there is also a possibility that there is no solution.