Consistency of Linear Equations using Rank and Determinants A system of linear equations can be considered consistent if there is at least one unique solutio...
A system of linear equations can be considered consistent if there is at least one unique solution for the variables. Two main concepts are used to determine the consistency of a linear system: rank and determinant.
Rank:
The rank of a matrix is a measure of linear independence of its columns.
If the rank of the coefficient matrix (also called the Jacobian) is equal to the number of equations in the system, then the system is consistent.
This means that there is exactly one solution for the variables, which is unique.
Determinant:
The determinant of a matrix is a scalar value that can be calculated from the matrix.
The determinant of the coefficient matrix is a single numerical value that can be calculated from the matrix.
The determinant of the coefficient matrix is zero if and only if the system is inconsistent.
A non-zero determinant indicates a consistent system, while a zero determinant indicates an inconsistent system.
Combined Approach:
To determine the consistency of a linear system using both rank and determinant, we perform the following steps:
Calculate the rank of the coefficient matrix.
Calculate the determinant of the coefficient matrix.
If the rank is equal to the number of equations and the determinant is not zero, the system is consistent.
If the rank is less than the number of equations, the system is inconsistent.
Example:
Consider the following system of linear equations:
x + y = 2
2x - y = 5
The rank of the coefficient matrix is 2, indicating that the system is consistent.
The determinant of the coefficient matrix is 3, which is not zero, indicating a consistent system.
Therefore, the system is consistent.
Conclusion:
The consistency of a linear system can be determined using both the rank and determinant concepts. A system is consistent if the rank of the coefficient matrix is equal to the number of equations and the determinant is not zero