Inverse of a matrix: statement, true or false?

[HelpNeeder]

If A is an n x n matrix and Ax = e_j is consistent for every j ∊ {1, 2, ..., n}, then A is invertible.

Could someone please help me understand why this statement is true/false?

 

[lex]

It is true. It implies that Ax=b has a solution for all b in [MATH]\mathbb{R^n}[/MATH],
which means A has a pivot in every row, which means A can be transformed into [MATH] \text{I}_n[/MATH] using e.r.o. s (since A is n x n, the pivots must be on the diagonal), therefore A is invertible.

In more detail:

Now (2) implies that A has a pivot in every row (by your previous work on Theorem 4):

A is n x n, so the pivots are on the diagonals, therefore it can be transformed into [MATH]\text{I}_n[/MATH] by elementary row operations.
Therefore A is invertible.