Answer
a.$Ax=0$ has a nontrivial solution.
b. $Ax=b$ has at least one solution for every possible b.
Work Step by Step
a) Because $A$ has two pivot positions and A is a 2 by 4 matrix, it has two free variables. 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 2 by 4 matrix, its columns span $\mathbb{R}^2$. For $Ax=b$ to have at least one solution for every possible b, A must have columns which span $\mathbb{R}^2$. Thus, $Ax=b$ has at least one solution for every possible b.
