## 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.