System of Equations Rank Calculator
Determine the rank of coefficient and augmented matrices to analyze system solvability.
Calculation Results
Rank of Coefficient Matrix (A):
Rank of Augmented Matrix ([A|b]):
System Solvability Analysis
The rank of a matrix indicates the number of linearly independent rows or columns. Comparing the rank of the coefficient matrix (A) and the augmented matrix ([A|b]) reveals the nature of solutions for the system of equations.
- Unique Solution: If rank(A) = rank([A|b]) = number of variables, the system has exactly one solution.
- Infinitely Many Solutions: If rank(A) = rank([A|b]) < number of variables, the system has infinitely many solutions.
- No Solution: If rank(A) < rank([A|b]), the system is inconsistent and has no solution.
Based on the calculated ranks:
Rank(A) = , Rank([A|b]) =
System Status:Unique SolutionInfinitely Many SolutionsNo Solution (Inconsistent System)
Understanding System of Equations and Rank
In linear algebra, a system of linear equations is a collection of equations involving variables in a linear manner. The rank of a matrix is a number that represents the maximum count of linearly independent rows or columns in the matrix. For a system of equations represented as Ax = b, where A is the coefficient matrix and b is the constant vector, analyzing the ranks of A and the augmented matrix [A|b] helps determine if the system has a solution and the nature of these solutions.
This calculator simplifies the process of finding the rank of these matrices, providing insights into whether a system of equations is solvable and if the solution is unique or infinite. Understanding these concepts is crucial in various fields like engineering, computer science, and economics for solving real-world problems modeled through linear systems.
For further learning, resources like Khan Academy's linear algebra section or MIT OpenCourseware on linear algebra can be very helpful.