A matrix is considered consistent when there exists at least one solution to the system of equations it represents. In simpler terms, if we can find a set of values for the variables that satisfy all the equations simultaneously, the matrix is deemed consistent.
To determine if a matrix is consistent, one typically looks at its augmented form, which includes the constant terms from the equations. By performing row reduction (Gaussian elimination), we can simplify the matrix and check for contradictions, such as a row that represents an impossibility (e.g., 0 = 1).
If the row reduction results in a row of zeros that corresponds to a zero constant (like 0 = 0), the matrix is consistent and may have either a unique solution or infinitely many solutions, depending on the number of free variables present.
In summary, the presence or absence of contradictions in the row-reduced form of the matrix directly determines its consistency:
- If the augmented matrix leads to a true statement like 0 = 0, it is consistent.
- If the augmented matrix leads to a false statement like 0 = 1, it is inconsistent.