Is it possible to have a system of equations that all equal 0, and not have each unknown's value be 0?

Question

I'm doing about a 2 hour long homework assignment where by hand I must construct a 10x10 matrix representing a system of equations. Based on the pattern I'm seeing, I can tell all of the equations are going to equal 0, but something tells me that is incorrect.

Can you have a system of equations of the form

and have x1 and x2 not equal 0?

Answer 1

Yes. This is actually one of the core topics in linear algebra. When the number of equations is at least the number of variables, it happens only when some of the equations are "redundant". For instance $x+y=0,2x+2y=0$. The second equation is redundant given the first, so there are nonzero solutions to this "system" namely any vector $\begin{bmatrix} t \\ -t \end{bmatrix}$ for $t \in \mathbb{R}$.

Answer 2

Suppose matrix $A$ is the one you propose above.
If we assume that a unique solution exists, we know that $A^{-1}$ exists. Then $$A\cdot x=b\implies x = A^{-1}\cdot b = 0\text{ when }b=0$$