What is the Condition for Consistency using Rank?

S7-SA2-0144

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

For a system of linear equations to have solutions (to be 'consistent'), the rank of the coefficient matrix must be equal to the rank of the augmented matrix. If these ranks are not equal, the system has no solutions.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya, and you're trying to figure out how many samosas and jalebis they bought. If Rohan's bill gives you one relationship between samosas and jalebis, and Priya's bill gives you another, for you to find a unique answer (or even any answer), both bills must 'agree' or be consistent. If one bill says '1 samosa + 1 jalebi = 20 rupees' and another says '1 samosa + 1 jalebi = 50 rupees', you know something is wrong – no solution is possible!

Worked Example

Step-by-Step

Let's check if the system of equations is consistent: x + y = 3 and 2x + 2y = 6.

1. Write the coefficient matrix (A) and the augmented matrix (A|B).
A = [[1, 1], [2, 2]]
A|B = [[1, 1, 3], [2, 2, 6]]
---2. Find the rank of matrix A using row operations.
R2 -> R2 - 2R1
A = [[1, 1], [0, 0]]
The number of non-zero rows is 1. So, Rank(A) = 1.
---3. Find the rank of the augmented matrix (A|B) using row operations.
R2 -> R2 - 2R1
A|B = [[1, 1, 3], [0, 0, 0]]
The number of non-zero rows is 1. So, Rank(A|B) = 1.
---4. Compare the ranks.
Rank(A) = 1 and Rank(A|B) = 1.
---5. Since Rank(A) = Rank(A|B), the system is consistent.
In this case, since the rank (1) is less than the number of variables (2), there are infinitely many solutions.

Answer: The system is consistent.

Why It Matters

Understanding consistency is crucial for solving real-world problems. Engineers use it to design stable structures, data scientists use it to train AI models that predict trends, and economists use it to model market behavior. This concept helps ensure that mathematical models have meaningful solutions.

Common Mistakes

MISTAKE: Not performing the same row operations on both the coefficient matrix and the augmented matrix when finding their ranks. | CORRECTION: Always ensure that when you're reducing the augmented matrix, the operations applied to the 'A' part are consistent with what you would do to 'A' alone.

MISTAKE: Incorrectly counting the number of non-zero rows after row reduction. | CORRECTION: A row is considered non-zero if it contains at least one non-zero element. Make sure all zeros are clearly identified.

MISTAKE: Confusing the condition for consistency with the condition for unique solutions or infinite solutions. | CORRECTION: Rank(A) = Rank(A|B) means 'consistent'. If this rank equals the number of variables, it's a unique solution. If this rank is less than the number of variables, it's infinite solutions.

Practice Questions

Try It Yourself

QUESTION: For the system x + y = 5, x + y = 7, find Rank(A) and Rank(A|B). Is the system consistent? | ANSWER: Rank(A) = 1, Rank(A|B) = 2. Not consistent.

QUESTION: Given the system: x + 2y + z = 4, 2x + 4y + 2z = 8, 3x + 6y + 3z = 12. Find the ranks of the coefficient and augmented matrices. Is the system consistent? If so, does it have unique or infinite solutions? | ANSWER: Rank(A) = 1, Rank(A|B) = 1. Consistent, with infinite solutions.

QUESTION: A small shop sells gulab jamuns (g) and rasgullas (r). On Monday, 2g + 3r = 100 rupees. On Tuesday, 4g + 5r = 180 rupees. Use the rank condition to determine if this system has a unique solution for the price of one gulab jamun and one rasgulla. | ANSWER: A = [[2, 3], [4, 5]], A|B = [[2, 3, 100], [4, 5, 180]]. After row operations, Rank(A) = 2, Rank(A|B) = 2. Since Rank(A) = Rank(A|B) = number of variables (2), there is a unique solution.

MCQ

Quick Quiz

If Rank(Coefficient Matrix) is NOT equal to Rank(Augmented Matrix) for a system of linear equations, what can be concluded?

The system has a unique solution.

The system has infinitely many solutions.

The system is inconsistent (no solution).

The system has trivial solutions.

The Correct Answer Is:

If the ranks are not equal, it means there's a contradiction in the equations, like 0 = 5, making the system inconsistent with no solution.

Real World Connection

In the Real World

When you use GPS on your phone to find the best route, multiple satellite signals create a system of equations. If these signals are inconsistent (e.g., due to interference or faulty data), your GPS might not be able to pinpoint your exact location or give you a valid route. Similarly, in ISRO's satellite launches, scientists ensure all parameters are consistent for a successful mission.

Key Vocabulary

Key Terms

Coefficient Matrix: A matrix formed by the coefficients of the variables in a system of linear equations. | Augmented Matrix: A matrix formed by adding the constant terms as an extra column to the coefficient matrix. | Rank of a Matrix: The maximum number of linearly independent row (or column) vectors in a matrix, or the number of non-zero rows after row reduction. | Consistent System: A system of linear equations that has at least one solution. | Inconsistent System: A system of linear equations that has no solution.

What's Next

What to Learn Next

Next, explore the conditions for unique solutions and infinitely many solutions based on rank. This will help you fully understand the different types of solutions a consistent system can have, building directly on what you've learned here.