What is the Application of Rank in Solving Linear Equations?

S7-SA2-0439

Grade Level:

Class 12

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

Definition

What is it?

The 'rank' of a matrix tells us how many independent equations we have in a system of linear equations. By comparing the rank of the original coefficient matrix with the rank of the augmented matrix (which includes the constant terms), we can determine if a system of equations has a unique solution, infinite solutions, or no solution at all.

Simple Example

Quick Example

Imagine you're trying to figure out the price of a samosa and a chai. If your friend tells you '2 samosas and 1 chai cost 50 rupees' and then '4 samosas and 2 chai cost 100 rupees', these are actually the same piece of information, just doubled. The 'rank' here would tell you you only have one truly independent piece of information, so you can't find exact prices for both samosa and chai individually, but many combinations are possible.

Worked Example

Step-by-Step

Let's check the consistency of the system: x + y = 5 and 2x + 2y = 10.

Step 1: Write the augmented matrix [A|B].
[ 1 1 | 5 ]
[ 2 2 | 10 ]

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Step 2: Apply row operations to reduce the matrix to row echelon form.
R2 -> R2 - 2R1
[ 1 1 | 5 ]
[ 0 0 | 0 ]

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Step 3: Find the rank of the coefficient matrix (A).
The matrix A is [ 1 1 ]
[ 0 0 ]
It has 1 non-zero row. So, rank(A) = 1.

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Step 4: Find the rank of the augmented matrix (A|B).
The matrix [A|B] is [ 1 1 | 5 ]
[ 0 0 | 0 ]
It has 1 non-zero row. So, rank(A|B) = 1.

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Step 5: Compare the ranks and the number of variables (n).
Here, rank(A) = 1, rank(A|B) = 1, and the number of variables (x, y) n = 2.
Since rank(A) = rank(A|B) < n, the system has infinitely many solutions.

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Answer: The system has infinitely many solutions.

Why It Matters

Understanding rank is crucial for engineers designing complex systems, like the electrical circuits in your mobile phone or the control systems in an EV. It helps AI/ML scientists build models that can solve problems efficiently. In finance, it helps analyze market data to predict trends, ensuring that the financial models are well-defined and solvable.

Common Mistakes

MISTAKE: Confusing the rank of the coefficient matrix with the rank of the augmented matrix. | CORRECTION: Always calculate both ranks separately and then compare them. The constant terms in the augmented matrix are very important.

MISTAKE: Incorrectly performing row operations, leading to wrong ranks. | CORRECTION: Double-check each row operation. Remember, you can multiply a row by a non-zero constant, add/subtract rows, or swap rows.

MISTAKE: Forgetting to compare the rank with the number of variables (n) when determining the type of solution. | CORRECTION: After finding rank(A) and rank(A|B), always compare them with 'n' (the number of unknowns) to decide between unique, infinite, or no solutions.

Practice Questions

Try It Yourself

QUESTION: For the system x + y = 3 and x + y = 5, what is rank(A) and rank(A|B)? | ANSWER: rank(A) = 1, rank(A|B) = 2

QUESTION: If rank(A) = 2, rank(A|B) = 2, and the number of variables is 2, how many solutions does the system have? | ANSWER: Unique solution

QUESTION: Determine if the system x + 2y = 7 and 3x + 6y = 21 has a unique, infinite, or no solution using ranks. | ANSWER: Infinitely many solutions

MCQ

Quick Quiz

If rank(A) = rank(A|B) = number of variables, what kind of solution does the system of linear equations have?

No solution

Unique solution

Infinitely many solutions

Cannot be determined

The Correct Answer Is:

When the rank of the coefficient matrix equals the rank of the augmented matrix, and both are equal to the number of variables, the system has a unique solution. This means there's only one specific set of values for the variables that satisfies all equations.

Real World Connection

In the Real World

Imagine an app like Google Maps or Ola that calculates the best route for your auto-rickshaw. It uses many linear equations to represent roads, traffic, and distances. The 'rank' concept helps the app quickly figure out if a path is possible (solution exists), if there are many equally good paths (infinite solutions), or if a destination is unreachable due to road closures (no solution). This ensures your ride is smooth and efficient.

Key Vocabulary

Key Terms

Matrix: A rectangular array of numbers arranged in rows and columns. | Rank: The maximum number of linearly independent rows or columns in a matrix. | Augmented Matrix: A matrix formed by combining the coefficient matrix and the constant terms of a system of linear equations. | Consistent System: A system of equations that has at least one solution (unique or infinite). | Inconsistent System: A system of equations that has no solution.

What's Next

What to Learn Next

Now that you understand how rank helps find solutions, you can explore Cramer's Rule and Gaussian Elimination. These methods are practical tools that use matrices to actually calculate the solutions for systems of linear equations, building directly on the consistency you've just learned about.