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What is the Consistency Check for Linear Systems using Rank Method?
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 Rank Method helps us figure out if a set of linear equations has a solution (consistent) or not (inconsistent). We do this by comparing the 'rank' of two special matrices formed from the equations.
Simple Example
Quick Example
Imagine you have three friends, and you know their total score in a game, and then some other combinations of their scores. Can you always find out each friend's individual score? The rank method helps us see if there's enough clear information to find a unique score for everyone, or if there are many possibilities, or even no way to find their scores at all!
Worked Example
Step-by-Step
Let's check if the system of equations x + y = 3 and 2x + 2y = 6 is consistent.
1. First, write the equations in matrix form AX = B. Here, A = [[1, 1], [2, 2]], X = [[x], [y]], and B = [[3], [6]].
---2. Form the coefficient matrix A: A = [[1, 1], [2, 2]].
---3. Form the augmented matrix [A|B]: [A|B] = [[1, 1, 3], [2, 2, 6]].
---4. Find the rank of matrix A. We can use row operations. R2 -> R2 - 2R1 gives [[1, 1], [0, 0]]. The number of non-zero rows is 1. So, rank(A) = 1.
---5. Find the rank of the augmented matrix [A|B]. R2 -> R2 - 2R1 gives [[1, 1, 3], [0, 0, 0]]. The number of non-zero rows is 1. So, rank([A|B]) = 1.
---6. Compare the ranks: rank(A) = 1 and rank([A|B]) = 1. Since rank(A) = rank([A|B]), the system is consistent. Also, since this rank (1) is less than the number of variables (2), it has infinitely many solutions.
Answer: The system is consistent and has infinitely many solutions.
Why It Matters
This method is super useful in fields like AI/ML for solving complex data problems, or in engineering to design stable structures. Scientists use it in climate science to model weather patterns and predict changes. Understanding this helps future engineers, data scientists, and researchers build better systems and solve real-world challenges.
Common Mistakes
MISTAKE: Not forming the augmented matrix correctly, mixing up coefficients and constants. | CORRECTION: Always ensure the constant terms (B) are added as the last column to the coefficient matrix (A) to form [A|B].
MISTAKE: Incorrectly performing row operations, leading to wrong ranks. | CORRECTION: Practice row operations carefully. Remember, you can multiply a row by a non-zero scalar, add a multiple of one row to another, or swap two rows.
MISTAKE: Forgetting to compare the rank with the number of variables. | CORRECTION: After finding rank(A) and rank([A|B]), remember to compare these ranks. If they are equal, also check if this common rank is equal to the number of variables to determine if it's a unique solution or infinitely many solutions.
Practice Questions
Try It Yourself
QUESTION: For the system x + y = 5, 2x + 2y = 10, find rank(A) and rank([A|B]). | ANSWER: rank(A) = 1, rank([A|B]) = 1
QUESTION: Determine if the system x + y = 2, x + y = 3 is consistent using the rank method. | ANSWER: Inconsistent (rank(A)=1, rank([A|B])=2)
QUESTION: For the system x + 2y + z = 4, 2x + y + 2z = 5, 3x + 3y + 3z = 10, determine if it is consistent and if it has a unique solution. | ANSWER: Inconsistent (rank(A)=2, rank([A|B])=3)
MCQ
Quick Quiz
If rank(A) = rank([A|B]) and this rank is equal to the number of variables, what kind of solution does the system have?
No solution
Infinitely many solutions
A unique solution
Two solutions
The Correct Answer Is:
C
When the rank of the coefficient matrix equals the rank of the augmented matrix, and this common rank equals the number of variables, the system has a unique solution. Options A and B describe other scenarios.
Real World Connection
In the Real World
Imagine you're developing a navigation app like Google Maps or Ola. To calculate the shortest route, the app solves many linear equations in real-time. The consistency check ensures that there's always a valid path (a solution) between your starting point and destination. If it were inconsistent, the app wouldn't be able to find a route!
Key Vocabulary
Key Terms
Linear System: A set of one or more linear equations involving the same set of variables. | Coefficient Matrix (A): A matrix containing only the coefficients of the variables in a linear system. | Augmented Matrix ([A|B]): A matrix formed by adding the constant terms (B) as an extra column to the coefficient matrix (A). | Rank of a Matrix: The maximum number of linearly independent row or column vectors in a matrix. | Consistent System: A system of equations that has at least one solution.
What's Next
What to Learn Next
Great job learning about consistency! Next, you can explore how to actually find these solutions using methods like Gaussian elimination or Cramer's Rule. This will help you not just know if a solution exists, but also what that solution actually is, which is crucial for practical problems.
