What would you have to know about the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution?

The Rouché-Capelli Theorem. This theorem establishes a connection between how a linear system behaves and the ranks of its coefficient matrix (A) and its counterpart the augmented matrix.

Then satisfying this theorem the system is consistent and has one single solution.

Explanation:

1) To answer that, you should have to know The Rouché-Capelli Theorem. This theorem establishes a connection between how a linear system behaves and the ranks of its coefficient matrix (A) and its counterpart the augmented matrix.

Then the system is consistent and has a unique solution.

E.g.

2) Writing it as Linear system

3) The Rank (A) is 3 found through Gauss elimination

4) The rank of (A|B) is also equal to 3, found through Gauss elimination:

So this linear system is consistent and has a unique solution.