How Row Operations Affect the Consistency of a Linear System | Explained with Examples

if a series of row operations on a linear system results in an inconsistent system, the original system is inconsistent

The statement you provided is not entirely accurate

The statement you provided is not entirely accurate. Let’s clarify and provide a correct statement:

If a series of row operations on a linear system results in a row of the form [0 0 0 … 0 | c] where c is a nonzero constant, then the original system is inconsistent.

To understand this concept, it is necessary to review some definitions and properties related to linear systems and row operations.

Linear System: A linear system consists of a set of equations involving variables and their coefficients. It can be represented in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

Inconsistent System: A linear system is considered inconsistent if there are no solutions that satisfy all the equations simultaneously. Geometrically, an inconsistent system represents a set of equations that do not define any common points, resulting in parallel or intersecting lines.

Row Operations: Row operations are transformations that can be applied to the rows of a matrix. They include:
1. Swapping: Interchanging two rows of a matrix.
2. Scaling: Multiplying a row by a nonzero scalar.
3. Row Addition: Adding a multiple of one row to another row.

Gaussian Elimination: Gaussian elimination is a systematic procedure of performing row operations to transform a linear system into reduced row-echelon form. This process simplifies the system and helps determine its consistency.

Consistent System: A linear system is considered consistent if there is at least one solution that satisfies all the equations simultaneously.

Now, let’s consider the implications of the statement mentioned in the question:

If a series of row operations on a linear system results in an inconsistent system, the original system is inconsistent.

This statement is incorrect because inconsistent systems can also be transformed into inconsistent systems through row operations. However, there are specific situations where row operations do guarantee consistency.

Row operations preserve the set of solutions of a linear system. Therefore, if a system is consistent, it will remain consistent after applying any number of row operations.

If a linear system is consistent, it means that there is at least one solution that satisfies all the equations simultaneously. In this case, it is still possible to perform row operations and obtain different forms of the same solution, but the consistency of the system remains unchanged.

However, if at any stage of performing row operations, a row of the form [0 0 0 … 0 | c] is obtained, where c is non-zero, the system becomes inconsistent. This happens because the row indicates an equation of the form 0 = c, which is impossible for any value of c.

To summarize, inconsistent systems can still remain inconsistent or become consistent through row operations, and consistent systems will always remain consistent. It is the presence of a row of the form [0 0 0 … 0 | c] with c non-zero that signifies inconsistency.

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