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What is the Condition for a Non-Homogeneous System to Have No Solution?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
A non-homogeneous system of linear equations has no solution when the equations contradict each other, making it impossible for any single set of values to satisfy all of them. This happens when the rank of the coefficient matrix is less than the rank of the augmented matrix.
Simple Example
Quick Example
Imagine you go to a snack shop. The owner tells you: '2 samosas and 1 chai cost Rs 50.' Then another person tells you: '2 samosas and 1 chai cost Rs 60.' These two statements can't both be true at the same time for the same items, right? This is like a system with no solution.
Worked Example
Step-by-Step
Let's check if this system has a solution:
Equation 1: x + y = 5
Equation 2: x + y = 7
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Step 1: Write the system in matrix form AX = B.
A = [[1, 1], [1, 1]] (Coefficient matrix)
X = [[x], [y]]
B = [[5], [7]]
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Step 2: Form the augmented matrix [A|B].
[A|B] = [[1, 1 | 5], [1, 1 | 7]]
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Step 3: Apply row operations to find the rank. Subtract Row 1 from Row 2 (R2 -> R2 - R1).
[A|B] = [[1, 1 | 5], [0, 0 | 2]]
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Step 4: Determine the rank of A and [A|B].
Rank of A (number of non-zero rows in A part) = 1 (from [[1, 1], [0, 0]])
Rank of [A|B] (number of non-zero rows in augmented matrix) = 2 (from [[1, 1 | 5], [0, 0 | 2]] because of the '2' in the last column)
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Step 5: Compare the ranks.
Since Rank(A) = 1 and Rank([A|B]) = 2, Rank(A) is NOT equal to Rank([A|B]).
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Answer: The system has no solution.
Why It Matters
Understanding when a system has no solution is crucial in fields like AI/ML for training models, and in engineering for designing stable structures. For example, in space technology, if equations describing rocket trajectories have no solution, it means the planned path is impossible, preventing costly errors.
Common Mistakes
MISTAKE: Assuming that if the determinant of the coefficient matrix is zero, there is always no solution. | CORRECTION: A zero determinant means there might be no solution OR infinitely many solutions. You need to check the rank of the augmented matrix to distinguish.
MISTAKE: Incorrectly performing row operations, especially when dealing with negative numbers or fractions, leading to wrong ranks. | CORRECTION: Double-check each step of your row operations. Practice with simple examples first to build accuracy.
MISTAKE: Confusing the rank of the coefficient matrix with the rank of the augmented matrix. | CORRECTION: Remember, the rank of the coefficient matrix (A) only considers the left part of the augmented matrix, while the rank of the augmented matrix ([A|B]) considers the entire matrix.
Practice Questions
Try It Yourself
QUESTION: For the system x + 2y = 3 and x + 2y = 5, does it have a solution? | ANSWER: No solution
QUESTION: Determine if the following system has a solution: x + y + z = 6, 2x + 2y + 2z = 10. | ANSWER: No solution
QUESTION: Find the value of 'k' for which the system 2x + 3y = 5 and 4x + 6y = k has no solution. | ANSWER: k is any real number except 10 (i.e., k ≠ 10)
MCQ
Quick Quiz
What is the condition for a non-homogeneous system of linear equations AX = B to have no solution?
Rank(A) = Rank([A|B])
Rank(A) < Rank([A|B])
Rank(A) > Rank([A|B])
Determinant(A) is non-zero
The Correct Answer Is:
B
A system has no solution when the rank of the coefficient matrix (A) is less than the rank of the augmented matrix ([A|B]). This means there's a contradiction in the equations.
Real World Connection
In the Real World
Imagine you are an engineer designing a new metro route in Delhi. You use linear equations to model traffic flow and resource allocation. If your equations for scheduling trains and available tracks lead to a 'no solution' scenario, it means the current plan is impossible to implement, and you need to rethink the design before construction begins.
Key Vocabulary
Key Terms
NON-HOMOGENEOUS SYSTEM: A system of linear equations where at least one constant term is not zero | COEFFICIENT MATRIX: A matrix containing only the coefficients of the variables | AUGMENTED MATRIX: A matrix formed by combining the coefficient matrix and the constant terms | RANK OF A MATRIX: The maximum number of linearly independent row (or column) vectors in a matrix
What's Next
What to Learn Next
Now that you understand when a system has no solution, you should explore the conditions for a non-homogeneous system to have a unique solution or infinitely many solutions. This will complete your understanding of all possible outcomes for such systems.
