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 ($\mathrm{\mid }A\mathrm{\mid }$) is non-zero, the system is consistent and has a unique solution.
• If $\mathrm{\mid }A\mathrm{\mid }=0$ and the determinant of the augmented matrix ($\mathrm{\mid }\text{aug}\left(A\right)\mathrm{\mid }$ formed by combining the coefficient matrix with the constant matrix is also $0$, the system is consistent but has infinitely many solutions.
• If $\mathrm{\mid }A\mathrm{\mid }=0$ but $\mathrm{\mid }\text{aug}\left(A\right)\mathrm{\mid }\mathrm{\ne }0$, 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:

$A=\left[\begin{array}{cc}2& 3\\ 4& 6\end{array}\right]$

1. Calculate the determinant of matrix $A$($\mathrm{\mid }A\mathrm{\mid }$).
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.