Consistency of the System of the Equations
Consistency of a System:
A system of linear equations can be classified into three main categories based on its consistency:
- A system of equations is consistent if it has at least one solution.
- If the system has a unique solution, it's called a uniquely consistent system.
- If the system has infinitely many solutions, it's termed a non-uniquely consistent system.
- An inconsistent system has no solution.
- It represents a situation where the equations contradict each other, leading to an impossibility of finding a common solution for all equations.
- If the determinant of the coefficient matrix () is non-zero, the system is consistent and has a unique solution.
- If and the determinant of the augmented matrix ( formed by combining the coefficient matrix with the constant matrix is also , the system is consistent but has infinitely many solutions.
- If but , the system is inconsistent.
Row Reduction or Gaussian Elimination:
- Row reduction to echelon or reduced echelon form can reveal the consistency of the system.
- A consistent system will have a solution space represented by parametric variables in reduced echelon form.
- An inconsistent system will reveal contradictions, such as where is a non-zero constant.
Consider a system of equations:
Using the coefficient matrix:
- Calculate the determinant of matrix ().
- Determine if the system is consistent or inconsistent based on the determinant.
- Optionally, perform row reduction to confirm the system's consistency and the nature of solutions.