Answer
Let the n$\times$n matrix $A$ be such that $A\mathrm{x}=0$ has only the trivial solution.
Then, there are no free variables in this equation (Th.2 of ch.1)
(all n columns contain a pivot)
Pivots are placed each in its own row, so the n rows of A also each contain a pivot.
In a row reduced echelon form of A, the pivots will be on the main diagonal,
meaning that A is row equivalent to $I_{n}.$
By Th.7, A is invertible.
Work Step by Step
The answer contains the explanation, as asked.
