I came across this simple exercise.
Suppose $A$ is $n\times n$ and the equation $A\textbf{x}=\textbf{0}$ has only the trivial solution. Explain why $A$ has $n$ pivot columns and $A$ is rowequivalent to $I_n$.
I understand that, because $A\textbf{x}=\textbf{0}$ implies $\textbf{x}=\textbf{0}$, $A$ must be invertible because $\bf{x} \mapsto A\bf{x}$ is one-to-one and if $A$ is invertible, it is obviously rowequivalent to $I_n$, but is there a more rigorous way to explain this?