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:

  1. Consistent System:

    • 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.
  2. Inconsistent 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.
  3. Determining Consistency:

    • Using Determinants:

      • If the determinant of the coefficient matrix (A) is non-zero, the system is consistent and has a unique solution.
      • If A=0 and the determinant of the augmented matrix (aug(A) formed by combining the coefficient matrix with the constant matrix is also 0, the system is consistent but has infinitely many solutions.
      • If A=0 but aug(A)0, 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 0=k where k is a non-zero constant.

Example:

Consider a system of equations:

2x+3y=10

4x+6y=20

Using the coefficient matrix:

A=[2346]

  1. Calculate the determinant of matrix A(A).
  2. Determine if the system is consistent or inconsistent based on the determinant.
  3. Optionally, perform row reduction to confirm the system's consistency and the nature of solutions.