## Reduction in Singular Systems

If we encounter a situation during the reduction step where
there is no remaining row that can be used to reduce the
current column, then the system of equations is singular:
that is, it does not have a unique solution.

- We can defer dealing with the singular situation
by swapping the (entire) column with a higher one to get
a non-zero element.
This is the usual approach.
- Since such a swap represents a
change in the order of the variables, the corresponding elements
in the variable (x) vector must also be swapped.
- Row and column swaps can also be combined
to get a non-zero pivot.

If no non-zero element remains that can be moved to the pivot
position, then we have come to the end of the line:
all the remaining rows are 0.

- The outcome depends on the constant vector.
- If all the zero matrix rows also have a zero constant term,
then the system is underdetermined, and an infinite number
of solutions exist (forming a linear subspace).
- If any constant term in a zero matrix row has a non-zero constant
term, then the system is inconsistent, and has no solutions.