Answer
a) $Ax=0$ has a nontrivial solution.
b) $Ax=b$ does not have at least one solution for every possible $b$.
Work Step by Step
a) Because $A$ has two pivot positions and $A$ is a 3 by 3 matrix, it has one free variable. To have a nontrivial solution to the equation $Ax=0$, $A$ must have a free variable. Thus, $Ax=0$ has a nontrivial solution.
b) Because $A$ has two pivot positions and $A$ is a 3 by 3 matrix, its columns do not span $\mathbb{R}^3$. For $Ax =b$ to have at least one solution for every possible $b$, $A$ must have columns which span $\mathbb{R}^3$. Thus, $Ax=b$ does not have at least one solution for every possible $b$.
