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What is the Consistency Check for Linear Systems using Determinants?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
The Consistency Check for Linear Systems using Determinants helps us figure out if a set of equations has a solution or not. It tells us if the equations 'agree' with each other (consistent) or 'disagree' (inconsistent). We use something called a 'determinant' to do this check.
Simple Example
Quick Example
Imagine you go to a chai stall. One day, you buy 2 samosas and 1 chai for Rs 50. Another day, you buy 4 samosas and 2 chai for Rs 100. Are these prices consistent? Using this check, we can see if the price of one samosa and one chai is fixed and makes sense for both purchases, or if something is off.
Worked Example
Step-by-Step
Let's check the consistency of these equations: x + y = 5 and 2x + 2y = 10.
Step 1: Write the equations in matrix form Ax = B. Here, A is [[1, 1], [2, 2]], x is [[x], [y]], and B is [[5], [10]].
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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 * 2) - (1 * 2) = 2 - 2 = 0.
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Step 3: Since det(A) = 0, we need to check the 'adjoint of A times B' (adj(A) * B). First, find adj(A). For a 2x2 matrix [[a, b], [c, d]], adj(A) = [[d, -b], [-c, a]]. So, adj(A) = [[2, -1], [-2, 1]].
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Step 4: Now, calculate adj(A) * B. adj(A) * B = [[2, -1], [-2, 1]] * [[5], [10]] = [[(2*5) + (-1*10)], [(-2*5) + (1*10)]] = [[10 - 10], [-10 + 10]] = [[0], [0]].
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Step 5: Since det(A) = 0 and adj(A) * B = 0, the system is consistent and has infinitely many solutions.
Answer: The system of equations is consistent and has infinitely many solutions.
Why It Matters
This check is super important in fields like AI/ML to solve complex problems, in engineering to design structures, and even in finance to model market trends. Engineers use it to ensure designs are stable, and data scientists use it to make sure their data models are reliable. Knowing this helps you build strong foundations for future careers in technology and science.
Common Mistakes
MISTAKE: Assuming a system is inconsistent just because det(A) = 0. | CORRECTION: If det(A) = 0, you MUST also check adj(A) * B. If adj(A) * B is not equal to the zero matrix, then it's inconsistent. If adj(A) * B is the zero matrix, it has infinitely many solutions.
MISTAKE: Forgetting to check the 'adjoint of A times B' (adj(A) * B) when det(A) is zero. | CORRECTION: The check for adj(A) * B is crucial when det(A) = 0 to distinguish between 'no solution' and 'infinitely many solutions'.
MISTAKE: Incorrectly calculating the determinant or the adjoint matrix. | CORRECTION: Double-check your calculations for det(A) and adj(A), especially the signs for the adjoint matrix elements.
Practice Questions
Try It Yourself
QUESTION: Is the system x + y = 3 and x - y = 1 consistent? | ANSWER: Yes, it is consistent (det(A) is not zero, so it has a unique solution).
QUESTION: Check the consistency for the system: x + 2y = 5 and 2x + 4y = 12. | ANSWER: Inconsistent (det(A) = 0, but adj(A) * B is not zero).
QUESTION: For the system 2x + 3y = 7 and 4x + 6y = 14, determine its consistency and the nature of its solutions. | ANSWER: Consistent, with infinitely many solutions (det(A) = 0 and adj(A) * B = 0).
MCQ
Quick Quiz
If for a system of linear equations Ax = B, the determinant of A (det(A)) is 0 and the product of the adjoint of A and B (adj(A) * B) is also the zero matrix, what can be said about the system?
It has a unique solution.
It is inconsistent (no solution).
It is consistent and has infinitely many solutions.
It cannot be determined without more information.
The Correct Answer Is:
C
When det(A) = 0 and adj(A) * B = 0, it means the equations are dependent and have countless solutions. Option A is for det(A) != 0, and Option B is for det(A) = 0 but adj(A) * B != 0.
Real World Connection
In the Real World
Imagine traffic lights in a big city like Mumbai. To ensure smooth traffic flow, engineers use linear equations to model traffic patterns. The consistency check helps them know if their proposed traffic light timings can actually work together (consistent) or if they will cause more jams (inconsistent). This prevents chaos on busy roads!
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 (either unique or infinitely many). | INCONSISTENT SYSTEM: A system of equations that has no solution. | ADJOINT MATRIX: A matrix formed by taking the transpose of the cofactor matrix.
What's Next
What to Learn Next
Next, you should explore 'Solving Linear Systems using Cramer's Rule'. This method directly uses determinants to find the actual values of x, y, and z when a unique solution exists, building directly on what you've learned here.
