Consider an augmented matrix of the form
\[\left [\begin{array}{c|c} A & \vec {b}\\ \end{array}\right ]\]
corresponding to a system of linear equations.

True or False?

1.
If there is more than one solution, \(\left [\begin{array}{c|c} A & \vec {b}\\ \end{array}\right ]\) has a row of zeros.
True False
2.
If \(\left [\begin{array}{c|c} A & \vec {b}\\ \end{array}\right ]\) has a row of zeros, there is more than one solution.
True False
3.
If there is no solution, \(\mbox {rref}(A)\) has a row of zeros.
True False
4.
If \(\mbox {rref}(A)\) has a row of zeros, there is no solution.
True False
5.
There is NO system that is inconsistent for every \(\vec {b}\).
True False
6.
If the system is consistent for some choice of \(\vec {b}\), it is consistent for every choice of \(\vec {b}\).
True False

Source

[Nicholson] W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2021, Open Edition, Problem 1.2.12.