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What is the Condition for a Non-Homogeneous System to Have Infinite Solutions?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
For a non-homogeneous system of linear equations to have infinite solutions, its coefficient matrix and augmented matrix must both have the same rank, and this rank must be less than the number of variables in the system. This means there's at least one free variable that can take any value, leading to endless possibilities.
Simple Example
Quick Example
Imagine you're buying 'chai' and 'samosa'. If the cost of 2 chai and 3 samosa is Rs 50, and also 4 chai and 6 samosa is Rs 100, these are essentially the same deal. You have two equations, but they tell you the same information. You can't find a unique price for chai and samosa; there are many combinations that work, like Rs 10 for chai and Rs 10 for samosa, or Rs 7 for chai and Rs 12 for samosa, and so on.
Worked Example
Step-by-Step
Consider the system of equations:
1. x + 2y + 3z = 6
2. 2x + 4y + 6z = 12
---Step 1: Write the augmented matrix [A|B].
[ 1 2 3 | 6 ]
[ 2 4 6 | 12 ]
---Step 2: Perform row operations to bring the matrix to row echelon form.
R2 = R2 - 2*R1
[ 1 2 3 | 6 ]
[ 0 0 0 | 0 ]
---Step 3: Determine the rank of the coefficient matrix (A) and the augmented matrix (A|B).
The coefficient matrix is [ 1 2 3; 0 0 0 ]. Its rank is 1 (one non-zero row).
The augmented matrix is [ 1 2 3 | 6; 0 0 0 | 0 ]. Its rank is 1 (one non-zero row).
---Step 4: Compare the ranks and the number of variables.
Rank(A) = 1
Rank(A|B) = 1
Number of variables (x, y, z) = 3
---Step 5: Check the condition for infinite solutions.
Since Rank(A) = Rank(A|B) = 1, and this rank (1) is less than the number of variables (3), the system has infinite solutions.
---Answer: The system has infinite solutions because the rank of the coefficient matrix equals the rank of the augmented matrix, and this rank is less than the number of variables.
Why It Matters
Understanding infinite solutions helps engineers design stable structures and computer scientists optimize algorithms. In AI/ML, it's crucial for training models where many parameter combinations can lead to good results. Knowing this helps you build smarter systems and solve complex real-world problems.
Common Mistakes
MISTAKE: Assuming that if Rank(A) = Rank(A|B), there will always be infinite solutions. | CORRECTION: While Rank(A) = Rank(A|B) is necessary, for infinite solutions, this common rank must also be LESS than the number of variables. If the rank equals the number of variables, there's a unique solution.
MISTAKE: Not reducing the matrix to its simplest row echelon form before determining the rank. | CORRECTION: Always perform all necessary row operations to get to row echelon form (or reduced row echelon form) to correctly count the number of non-zero rows, which gives the true rank.
MISTAKE: Confusing the number of equations with the number of variables when checking the condition. | CORRECTION: The condition compares the rank with the NUMBER OF VARIABLES, not the number of equations. A system can have more equations than variables, or vice-versa.
Practice Questions
Try It Yourself
QUESTION: For a non-homogeneous system Ax=B, if Rank(A) = Rank(A|B) = 2 and there are 3 variables, how many solutions does it have? | ANSWER: Infinite solutions.
QUESTION: Consider the system: x + y = 5; 2x + 2y = 10. Does this system have infinite solutions? Explain why. | ANSWER: Yes, it has infinite solutions. The augmented matrix is [[1 1 | 5], [2 2 | 10]]. Reducing it gives [[1 1 | 5], [0 0 | 0]]. Rank(A) = 1, Rank(A|B) = 1. Number of variables = 2. Since Rank(A) = Rank(A|B) < Number of variables, there are infinite solutions.
QUESTION: If a system of 3 equations with 3 variables has Rank(A) = 3 and Rank(A|B) = 3, what kind of solution does it have? | ANSWER: Unique solution. (Because Rank(A) = Rank(A|B) = Number of variables).
MCQ
Quick Quiz
What is the primary condition for a non-homogeneous system of linear equations to have infinite solutions?
Rank of coefficient matrix equals rank of augmented matrix.
Rank of coefficient matrix is less than the number of variables.
Rank of coefficient matrix equals rank of augmented matrix, and this rank is less than the number of variables.
Rank of coefficient matrix is greater than the rank of augmented matrix.
The Correct Answer Is:
C
Option C correctly states both necessary conditions: the ranks must be equal, AND that common rank must be less than the number of variables. Options A and B alone are insufficient.
Real World Connection
In the Real World
Imagine you are building a smart traffic light system in Mumbai. You have sensors collecting data about vehicle density (variables) at different intersections (equations). If your equations are redundant (like saying 'two cars pass in 5 minutes' and then 'four cars pass in 10 minutes' for the same spot), the system will find infinite ways to manage traffic, making it hard to predict the best flow. This concept helps engineers design systems with unique, predictable solutions for optimal traffic management, or understand when a system is too flexible.
Key Vocabulary
Key Terms
Non-homogeneous system: A system of linear equations where at least one constant term (on the right side) is not zero. | Coefficient matrix: A matrix formed by the coefficients of the variables in a system of linear equations. | Augmented matrix: A matrix formed by combining the coefficient matrix with the constant terms of the equations. | Rank of a matrix: The maximum number of linearly independent row or column vectors in the matrix. | Variables: The unknown quantities (like x, y, z) that we are trying to find in the equations.
What's Next
What to Learn Next
Now that you understand infinite solutions, explore what happens when a system has 'no solution' or a 'unique solution'. This will complete your understanding of the different types of solutions for linear systems, which is super important for higher-level math and science!
