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What is the Consistency of a System using Rank?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The consistency of a system using rank tells us if a set of linear equations has a solution or not. If a system is consistent, it means we can find values for the variables that satisfy all equations. If it's inconsistent, no such solution exists.
Simple Example
Quick Example
Imagine you go to a shop. The owner says, '2 samosas and 1 chai cost 50 rupees.' Then he says, '4 samosas and 2 chai cost 100 rupees.' These two statements are consistent because the second is just double the first. But if he said, '4 samosas and 2 chai cost 110 rupees,' it would be inconsistent because it contradicts the first statement.
Worked Example
Step-by-Step
Let's check the consistency of this system: x + y = 5 and 2x + 2y = 10.
1. First, write the augmented matrix [A|B].
A = [[1, 1], [2, 2]] and B = [[5], [10]].
So, [A|B] = [[1, 1, | 5], [2, 2, | 10]]
2. Reduce the matrix to its row echelon form. Perform R2 -> R2 - 2R1.
[[1, 1, | 5], [2 - 2*1, 2 - 2*1, | 10 - 2*5]]
[[1, 1, | 5], [0, 0, | 0]]
3. Calculate the rank of matrix A (rank(A)). The number of non-zero rows in A is 1. So, rank(A) = 1.
4. Calculate the rank of the augmented matrix [A|B] (rank([A|B])). The number of non-zero rows in [A|B] is 1. So, rank([A|B]) = 1.
5. Compare the ranks. Here, rank(A) = rank([A|B]) = 1.
6. Since the ranks are equal, the system is consistent. Also, since the rank (1) is less than the number of variables (2, which are x and y), it has infinitely many solutions.
Answer: The system is consistent and has infinitely many solutions.
Why It Matters
Understanding consistency is crucial in many fields! In AI/ML, it helps ensure that data models have valid solutions. In Engineering, it ensures that designs for bridges or circuits are physically possible. Doctors use it to check if different drug dosages combine consistently for a patient.
Common Mistakes
MISTAKE: Not reducing the matrix to its simplest row echelon form before finding the rank. | CORRECTION: Always ensure the matrix is in row echelon form (or reduced row echelon form) where leading entries are 1s and columns below them are zeros, before counting non-zero rows for rank.
MISTAKE: Confusing the rank of the coefficient matrix (A) with the rank of the augmented matrix ([A|B]). | CORRECTION: Remember to calculate both ranks separately. rank(A) is based only on the coefficients of variables, while rank([A|B]) includes the constant terms as well.
MISTAKE: Assuming consistency automatically means a unique solution. | CORRECTION: Consistency means a solution exists. If rank(A) = rank([A|B]) = number of variables, then there's a unique solution. If rank(A) = rank([A|B]) < number of variables, there are infinitely many solutions.
Practice Questions
Try It Yourself
QUESTION: For the system x + y = 3 and x + y = 5, find rank(A) and rank([A|B]). Is the system consistent? | ANSWER: rank(A) = 1, rank([A|B]) = 2. The system is inconsistent.
QUESTION: Determine the consistency of the system: x + y + z = 6, 2y + 5z = -4, 2x + 5y - z = 27. | ANSWER: The system is consistent and has a unique solution. (rank(A) = 3, rank([A|B]) = 3, number of variables = 3)
QUESTION: For what value of 'k' will the system x + 2y = 5 and 3x + ky = 15 be consistent? | ANSWER: k = 6. (For consistency, the rows must be linearly dependent if the rank is less than the number of variables, or the system must have a unique solution. If k=6, the second equation is 3 times the first, leading to rank(A)=1, rank([A|B])=1, consistent with infinite solutions.)
MCQ
Quick Quiz
If for a system of linear equations, rank(A) is not equal to rank([A|B]), what can you conclude about the system?
It has a unique solution.
It has infinitely many solutions.
It is inconsistent and has no solution.
It is consistent but the solution cannot be found.
The Correct Answer Is:
C
If rank(A) is not equal to rank([A|B]), it means there's a contradiction in the equations, making the system inconsistent with no possible solution.
Real World Connection
In the Real World
When you use a navigation app like Google Maps or Ola/Uber, it constantly solves complex systems of equations to find the best route. If the underlying data (like road closures, traffic, speed limits) creates an inconsistent system, the app wouldn't be able to suggest a valid path, or might even crash. This concept helps ensure the app can always find a consistent route.
Key Vocabulary
Key Terms
RANK: The maximum number of linearly independent rows or columns in a matrix. | COEFFICIENT MATRIX (A): A matrix containing only the coefficients of the variables. | AUGMENTED MATRIX ([A|B]): A matrix formed by combining the coefficient matrix (A) with the constant terms (B). | CONSISTENT SYSTEM: A system of equations that has at least one solution. | INCONSISTENT SYSTEM: A system of equations that has no solution.
What's Next
What to Learn Next
Great job understanding consistency! Next, you should explore how to find the actual solutions for consistent systems using techniques like Gaussian elimination or Cramer's Rule. This builds on what you've learned and helps you solve real-world problems more effectively.
