This is another question from Anton, Elementary Linear Algebra 9th Ed.
The first system is:
$\begin{array}{1}x_{1}+3x_{2}-x_{3}=0\\
\,\,\qquad x_{2}-8x_{3}=0\\
\,\,\,\,\qquad\qquad 4x_{3}=0\end{array}$
Correction as demonstrated in the comments below. This system does not have a nontrivial solution.
The second system is:
$\begin{array}{1}a_{11}x_{1}+a_{12}x_{2}+a_{13}x_{3}=0\\
a_{21}x_{1}+a_{22}x_{2}+a_{23}x_{3}=0\end{array}$
For this second system it is an underdetermined homogeneous system of linear equations and so it has infinite solutions.
The third system is:
$\begin{array}{1}3x_{1}-2x_{2}=0\\
6x_{1}-4x_{2}=0\end{array}$
This system is dependent linear equations, so they are collinear with infinite solutions.
The fourth system is:
$\begin{array}{1}2x_{1}-3x_{2}+4x_{3}-x_{4}=0\\
7x_{1}+x_{2}-8x_{3}+9x_{4}=0\\
2x_{1}+8x_{2}+x_{3}-x_{4}=0\end{array}$
With this fourth one, I cannot see any obvious sign to tell me there is more than the trivial solution. Is there something here that I am missing. All I can do at a glance is identify that it is not any of the other three cases. Maybe row-reduction might discover something else but this question is not supposed to use pencil and paper, i.e. no computation.
Thanks.
Best Answer
Hint:-
If you are given m homogeneous equations in n unknowns and $m < n$, then you always get infinitely many solutions, since at least one variable becomes free.
For a homogeneous system, if you have m equations in n unknowns with $m = n$, then you get only the trivial solution.
I hope this helps you solve the problems.