What is the Condition for Consistency using Determinants?

S7-SA2-0143

Grade Level:

Class 12

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

Definition

What is it?

The condition for consistency using determinants tells us if a system of linear equations has a solution or not. A system is 'consistent' if it has at least one solution (either a unique solution or infinitely many solutions). We use determinants to check this condition without actually solving the equations.

Simple Example

Quick Example

Imagine you have two friends, Rahul and Priya, who bought some sweets. Rahul bought 2 laddoos and 1 barfi for 70 rupees. Priya bought 4 laddoos and 2 barfis for 140 rupees. If we write this as equations, we can use determinants to quickly see if their sweet prices make sense (are 'consistent') or if there's a trick, without finding the price of each sweet.

Worked Example

Step-by-Step

Let's check the consistency of the system:
2x + 3y = 5
4x + 6y = 10

Step 1: Write the equations in matrix form AX = B.
A = [[2, 3], [4, 6]], X = [[x], [y]], B = [[5], [10]]
---Step 2: Calculate the determinant of the coefficient matrix A, denoted as det(A).
det(A) = (2 * 6) - (3 * 4) = 12 - 12 = 0
---Step 3: Since det(A) = 0, we need to check the determinants of the adjoint matrix times B, or in simpler terms, check the consistency further. For a 2x2 system, if det(A) = 0, we check if the equations are proportional. Here, the second equation (4x + 6y = 10) is exactly twice the first equation (2x + 3y = 5). This means they represent the same line.
---Step 4: If det(A) = 0 and the equations are proportional (or the adjoint(A) * B = 0 for higher order systems), the system has infinitely many solutions. This means the system is consistent.

Answer: The system is consistent and has infinitely many solutions.

Why It Matters

Understanding consistency helps engineers design stable structures, data scientists train accurate AI models, and economists predict market trends. For example, in AI, if a system of equations defining a model is inconsistent, the model won't learn properly. Engineers use this in designing circuits and solving problems in robotics.

Common Mistakes

MISTAKE: Confusing 'consistent' with 'unique solution' | CORRECTION: A consistent system can have a unique solution OR infinitely many solutions. Inconsistent means no solution.

MISTAKE: Assuming det(A) = 0 always means no solution | CORRECTION: If det(A) = 0, it means either infinitely many solutions (if the system is dependent) or no solution (if the system is contradictory). You need to check further.

MISTAKE: Incorrectly calculating the determinant, especially for 3x3 matrices | CORRECTION: Double-check your calculations for cofactors and minor determinants to avoid sign errors and arithmetic mistakes.

Practice Questions

Try It Yourself

QUESTION: Is the system consistent: x + y = 3, x + y = 5? | ANSWER: Inconsistent

QUESTION: For the system 3x + 2y = 7, 6x + 4y = 14, calculate det(A) and state if it's consistent. | ANSWER: det(A) = 0. The system is consistent (infinitely many solutions).

QUESTION: A shopkeeper sells two types of snacks, Samosa (x) and Jalebi (y). On Monday, he sold 3 samosas and 2 jalebis for Rs 90. On Tuesday, he sold 6 samosas and 4 jalebis for Rs 150. Is this pricing consistent? Use determinants to check. | ANSWER: det(A) = 0. The system is inconsistent (because 6x+4y=180 would be consistent, but 6x+4y=150 is not).

MCQ

Quick Quiz

If for a system of linear equations AX = B, the determinant of the coefficient matrix A, i.e., det(A), is non-zero, what does this imply about the system?

The system is inconsistent

The system has infinitely many solutions

The system has a unique solution and is consistent

The system has no solution

The Correct Answer Is:

If det(A) is not zero, it means the inverse of A exists, and we can find a unique solution X = A^-1 * B. This implies the system is consistent and has exactly one unique solution.

Real World Connection

In the Real World

When you use GPS on your phone to find the best route, it solves many equations to pinpoint your location and calculate distances. If the system of equations were inconsistent, your GPS would show you in the wrong place or give impossible routes! Similarly, ISRO scientists use these principles to track satellites and ensure their paths are consistent with physical laws.

Key Vocabulary

Key Terms

DETERMINANT: A special number calculated from a square matrix. | CONSISTENT SYSTEM: A system of equations with at least one solution. | INCONSISTENT SYSTEM: A system of equations with no solution. | UNIQUE SOLUTION: Exactly one specific set of values that satisfies all equations. | INFINITELY MANY SOLUTIONS: An unlimited number of solutions that satisfy all equations.

What's Next

What to Learn Next

Next, you should explore 'Cramer's Rule' and 'Matrix Inverse Method'. These methods actually help you find the solutions for consistent systems, building directly on your understanding of determinants and consistency.