Consistency of Linear System of Equations A linear system of n equations with n unknowns can be represented by the following matrix equation: $$\begin{bmatri...
A linear system of n equations with n unknowns can be represented by the following matrix equation:
where:
a is the coefficient matrix
x is the vector of unknowns
b is the vector of constants
The consistency of this linear system can be determined by examining the rank of the coefficient matrix.
Rank(a) = n: If the rank of the coefficient matrix (n) is equal to the number of equations (n), then the linear system is consistent. This means that there is exactly one unique solution to the system.
Rank(a) < n: If the rank of the coefficient matrix is less than n, then the linear system is inconsistent. This means that there is no unique solution to the system.
Rank(a) = n and b ≠0: If the rank of the coefficient matrix is n and the vector b is not the zero vector, then the linear system is consistent. This means that there is exactly one unique solution to the system.
Additional notes:
A consistent linear system has unique solutions, meaning the solution to the system is determined uniquely.
An inconsistent system has no unique solution, meaning the solution is not determined uniquely.
A system with a rank of n is equivalent to a set of n linearly independent equations.
A system with n equations and n unknowns can be considered under a single matrix