The Impact of Row Operations on the Consistency of Linear Systems

if a row operation is done to a consistent linear system, the resulting system must be consistent.

Yes, if a row operation is performed on a consistent linear system, the resulting system must also be consistent

Yes, if a row operation is performed on a consistent linear system, the resulting system must also be consistent.

In linear algebra, a consistent linear system refers to a system of equations where there exists at least one solution. On the other hand, an inconsistent system has no solution.

Row operations are manipulations that are performed on the equations of a system to simplify it or to transform it into an equivalent system. The three basic row operations are:

1. Interchanging two equations.
2. Multiplying an equation by a non-zero constant.
3. Adding or subtracting a multiple of one equation to or from another equation.

Importantly, all of these row operations are reversible. That is, for every row operation performed, there is an inverse operation that can undo it.

When a row operation is applied to a consistent linear system, it does not change the solutions of the system. This is because the fundamental properties of linear equations are preserved during row operations. Consequently, if the original system had at least one solution, the resulting system will still have at least one solution. Hence, the resulting system is also consistent.

It is worth noting that while row operations preserve consistency, they may change the form of the system or its solutions. Therefore, row operations are often used to simplify a system or to find a more convenient form for solving it.

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