It is true. It implies that A

**x**=

**b** has a solution for all

**b** in \(\displaystyle \mathbb{R^n}\),

which means A has a pivot in every row, which means A can be transformed into \(\displaystyle \text{I}_n\) 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 \(\displaystyle \text{I}_n\) by elementary row operations.

Therefore A is invertible.