Answer
a. $Ax=0$ does not have 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 2 matrix, it has no free variables. To have a nontrivial solution to the equation $Ax=0$, A must have a free variable. Thus, $Ax=0$ does not have a nontrivial solution.
b) Because A has two pivot positions and A is a 3 by 2 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 $b$.
