What is System Consistency using Determinants? | Simple Explanation for Class 10

S6-SA1-0388

What is the Consistency of a System of Linear Equations using Determinants?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The consistency of a system of linear equations tells us if there are any solutions (values for the variables that make all equations true). Using determinants helps us quickly figure out if a system has a unique solution, no solution, or infinitely many solutions, without actually solving for the variables.

Simple Example

Quick Example

Imagine you buy two different kinds of sweets, ladoos and barfis, from two shops. Shop 1 says: '2 ladoos + 3 barfis cost Rs 100'. Shop 2 says: '4 ladoos + 6 barfis cost Rs 200'. Are these prices consistent? A determinant check would quickly tell us if there's a unique price for each sweet, or if the information is redundant (infinitely many solutions), or contradictory (no solution).

Worked Example

Step-by-Step

Let's check the consistency for this system: x + 2y = 5 and 2x + 4y = 10.

Step 1: Write the equations in matrix form AX = B. Here, A is [[1, 2], [2, 4]], X is [[x], [y]], and B is [[5], [10]].
---Step 2: Calculate the determinant of matrix A, denoted as det(A). For a 2x2 matrix [[a, b], [c, d]], det(A) = ad - bc. So, det(A) = (1 * 4) - (2 * 2) = 4 - 4 = 0.
---Step 3: Since det(A) = 0, the system either has no solution or infinitely many solutions. We need to check the adjoint matrix and B.
---Step 4: Calculate the adjoint of A, denoted as adj(A). For [[a, b], [c, d]], adj(A) = [[d, -b], [-c, a]]. So, adj(A) = [[4, -2], [-2, 1]].
---Step 5: Calculate adj(A) * B. This is [[4, -2], [-2, 1]] * [[5], [10]] = [[(4*5) + (-2*10)], [(-2*5) + (1*10)]] = [[20 - 20], [-10 + 10]] = [[0], [0]].
---Step 6: Since det(A) = 0 AND adj(A) * B = 0, the system has infinitely many solutions.
---Answer: The system is consistent with infinitely many solutions.

Why It Matters

Understanding consistency helps engineers design stable structures and AI algorithms make reliable predictions. In fields like space technology, checking consistency ensures that rocket trajectories or satellite positions can be uniquely determined, preventing costly errors. This concept is fundamental for anyone working with data and systems in the future.

Common Mistakes

MISTAKE: Assuming det(A) = 0 always means no solution. | CORRECTION: If det(A) = 0, you MUST also check adj(A) * B. If adj(A) * B is also 0, there are infinitely many solutions. If adj(A) * B is not 0, then there is no solution.

MISTAKE: Incorrectly calculating the determinant, especially for 3x3 matrices. | CORRECTION: Double-check your determinant calculation. Remember the sign changes for cofactors (+ - + for the first row).

MISTAKE: Confusing 'consistent' with 'unique solution'. | CORRECTION: A system is 'consistent' if it has AT LEAST ONE solution (either unique or infinitely many). It is 'inconsistent' only if it has NO solution.

Practice Questions

Try It Yourself

QUESTION: For the system x + y = 3 and 2x + 2y = 7, find det(A) and determine its consistency. | ANSWER: det(A) = 0. adj(A)*B is not 0. The system is inconsistent (no solution).

QUESTION: Determine the consistency of the system: 3x - y = 5 and 6x - 2y = 10. | ANSWER: det(A) = 0. adj(A)*B is 0. The system is consistent with infinitely many solutions.

QUESTION: For what value of 'k' will the system 2x + 3y = 5 and 4x + ky = 10 have infinitely many solutions? | ANSWER: k = 6

MCQ

Quick Quiz

If for a system of linear equations AX = B, det(A) is not equal to 0, what can we say about its consistency?

It is inconsistent (no solution).

It is consistent with infinitely many solutions.

It is consistent with a unique solution.

More information is needed to determine consistency.

The Correct Answer Is:

If det(A) is not equal to 0, it guarantees that the inverse of A exists, meaning there is exactly one unique solution (X = A_inverse * B). Therefore, the system is consistent with a unique solution.

Real World Connection

In the Real World

Imagine a logistics company like Delhivery or Ecom Express planning delivery routes. They use systems of linear equations to optimize fuel usage and delivery times. Checking consistency helps them quickly see if their planning equations have a clear, single best route (unique solution), many possible good routes (infinitely many), or if their parameters conflict, making a route impossible (no solution).

Key Vocabulary

Key Terms

DETERMINANT: A special number calculated from a square matrix that tells us properties of the matrix. | CONSISTENT SYSTEM: A system of equations that has at least one solution. | INCONSISTENT SYSTEM: A system of equations that has no solution. | UNIQUE SOLUTION: Exactly one set of values for the variables that satisfies all equations. | INFINITELY MANY SOLUTIONS: An unlimited number of sets of values that satisfy all equations.

What's Next

What to Learn Next

Now that you understand consistency, you can explore Cramer's Rule, which uses determinants to actually find the unique solutions for consistent systems. This will further strengthen your foundation for advanced topics in linear algebra and its applications in technology!