Rank of Matrix: The rank of a matrix is the dimension of the vector space generated by its columns. It is the maximum number of linearly independent column vectors in the matrix.
Augmented Matrix (A|B): A matrix that includes both the coefficients and the constant terms (B) of a system of linear equations.
Relationship Between Rank of A and Rank of Augmented Matrix (A|B)
Rank of A = Rank of A|B:
B is a linear combination of the columns of A.
The solution to the system of equations exists.
The solution could be either unique or infinite.
Rank of A ≠ Rank of A|B:
B is not a linear combination of the columns of A.
It implies that adding B increases the rank by at most 1.
The solution to the system of equations does not exist.
Scenarios and Inferences
If a new vector B is added to A:
B is linearly dependent on vectors in A: Rank does not change (Rank of A = Rank of A|B).
B is linearly independent: Rank increases by 1 (Rank of A|B = Rank of A + 1).
In terms of the solution:
If ranks are equal: The system has a solution, either unique or infinite.
If ranks are not equal: There is no solution.
Zero and Non-Zero Concept
Rank of A = Rank of A|B implies no extra pivots are added; no non-zero rows are added beyond the row echelon form.
Rank of A ≠ Rank of A|B implies an extra pivot is added; a non-zero row appears where previously there was none.
Understanding Pivots
Pivot Element: A non-zero element in the matrix that is used to eliminate other elements to achieve row echelon form.
If adding B adds a pivot, it changes the rank, implying a new vector was introduced.
Free Variables and Rank
Free Variable Exists: Rank of A < number of columns (n).
No Free Variable: Rank of A = n, meaning each column has a pivot.
Conclusion
The difference or equality in ranks of A and A|B can determine the existence of solutions and the nature of the system (consistent or inconsistent).
The free variable concept can be directly linked to rank conditions.
Flowchart Insights
Inconsistent Solution: Rank of A ≠ Rank of A|B or 0,0,0, non-zero exists after Gaussian elimination.
Consistent Solution: Rank of A = Rank of A|B.
Check if there are free variables (Rank < n) for infinite solutions.
No free variables (Rank = n) implies a unique solution.
Additional Notes
After Gaussian elimination, verify the pivot positions to determine the rank and thereby infer about the solutions.