I'm studying Linear Algebra for the second time, using Hoffmann & Kunze. Currently I'm trying to prove the following theorem:
Theorem 7. If $A$ is an $n \times n$ matrix, then $A$ is row-equivalent to the $n \times n$ identity matrix if and only if the system of equations $A\vec{x} = \vec{0}$ has only the trivial solution $\vec{x} = \vec{0}$.
The given proof in the textbook is, I find, obscure, and I don't understand it at all, so I came up with my own proof. However, it depends on $det(A) \neq 0$ and I'm not sure if this assumption can be made.
I rewrote the theorem as:
Let $A$ be a square matrix. Prove that $A$ ~ $I_n$ if and only if $A\vec{x} = \vec{0}$ has only the trivial solution.
My proof:
Let $det(A) \neq 0$. Then $rref(A) = I_n$. If $A\vec{x} = \vec{0}$, then $det(A\vec{x}) = 0 \rightarrow \vec{x} = \vec{0}$, so $\vec{0}$ is indeed the only solution.
The problem with my proof is, even if it's correct, the authors only introduce determinants around page $200$ and this theorem is from chapter $1$. So really I'm looking for a proof that does not rely upon determinants.