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:

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 nonuniquely consistent system.

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.

Determining Consistency:

Using Determinants:
 If the determinant of the coefficient matrix ($\mathrm{\mid}A\mathrm{\mid}$) is nonzero, the system is consistent and has a unique solution.
 If $\mathrm{\mid}A\mathrm{\mid}=0$ and the determinant of the augmented matrix ($\mathrm{\mid}\text{aug}(A)\mathrm{\mid}$ 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 $\mathrm{\mid}\text{aug}(A)\mathrm{\mid}\mathrm{\ne}0$, 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 nonzero constant.

Example:
Consider a system of equations:
$2x+3y=10$
$4x+6y=20$
Using the coefficient matrix:
$A=\left[\begin{array}{cc}{\textstyle 2}& {\textstyle 3}\\ {\textstyle 4}& {\textstyle 6}\end{array}\right]$
 Calculate the determinant of matrix $A$($\mathrm{\mid}A\mathrm{\mid}$).
 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.