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If the coefficient matrix is Cn×n and its corresponding rank is c also the augmented matrix is An×(n+1) and its corresponding rank is a, then how many statements given below are incorrect?
I. If c ≠ a, the equations are inconsistent with an infinite number of solutions.
II. If c = a < n, the equations are inconsistent
III. If c = a =n, equation may be inconsistent.
Correct answer is '2'. Can you explain this answer?
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Verified Answer
If the coefficient matrix isCn×n and its corresponding rank isca...
- If c ≠ a, the equations are inconsistent i.e. no solution possible
- If c = a < n, the equations are consistent with infinite number of solutions
- If c = a =n, the equations are consistent and there is a unique solution.
Most Upvoted Answer
If the coefficient matrix isCn×n and its corresponding rank isca...
Understanding the Statements
To analyze the correctness of the statements regarding the rank of matrices and the consistency of equations, let’s break down each statement.
Statement I: If c ≠ a, the equations are inconsistent with an infinite number of solutions.
- This statement is incorrect. If the rank of the coefficient matrix (c) is less than the rank of the augmented matrix (a), it indicates inconsistency, meaning there are no solutions, not an infinite number.
Statement II: If c = a < n,="" the="" equations="" are="" />
- This statement is also incorrect. If the rank of both the coefficient and augmented matrices are equal (c = a) and less than the number of variables (n), it implies there are infinitely many solutions, not inconsistency.
Statement III: If c = a = n, the equation may be inconsistent.
- This statement is correct. When the ranks are equal to the number of variables, it indicates a unique solution. However, if the equations are dependent or inconsistent, it may also lead to inconsistency in certain cases.
Conclusion
- Therefore, out of the three statements, two are incorrect (Statements I and II).
- This aligns with the correct answer of '2' incorrect statements.
Understanding the rank and consistency of equations is crucial in linear algebra, especially in Computer Science Engineering.
To analyze the correctness of the statements regarding the rank of matrices and the consistency of equations, let’s break down each statement.
Statement I: If c ≠ a, the equations are inconsistent with an infinite number of solutions.
- This statement is incorrect. If the rank of the coefficient matrix (c) is less than the rank of the augmented matrix (a), it indicates inconsistency, meaning there are no solutions, not an infinite number.
Statement II: If c = a < n,="" the="" equations="" are="" />
- This statement is also incorrect. If the rank of both the coefficient and augmented matrices are equal (c = a) and less than the number of variables (n), it implies there are infinitely many solutions, not inconsistency.
Statement III: If c = a = n, the equation may be inconsistent.
- This statement is correct. When the ranks are equal to the number of variables, it indicates a unique solution. However, if the equations are dependent or inconsistent, it may also lead to inconsistency in certain cases.
Conclusion
- Therefore, out of the three statements, two are incorrect (Statements I and II).
- This aligns with the correct answer of '2' incorrect statements.
Understanding the rank and consistency of equations is crucial in linear algebra, especially in Computer Science Engineering.
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If the coefficient matrix isCn×n and its corresponding rank iscalso the augmented matrix isAn×(n+1) and its corresponding rank isa,then how many statements given below are incorrect?I. Ifc ≠ a, the equations are inconsistent with an infinite number of solutions.II. Ifc=a < n, the equations are inconsistentIII. Ifc=a =n, equation may be inconsistent.Correct answer is '2'. Can you explain this answer?
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If the coefficient matrix isCn×n and its corresponding rank iscalso the augmented matrix isAn×(n+1) and its corresponding rank isa,then how many statements given below are incorrect?I. Ifc ≠ a, the equations are inconsistent with an infinite number of solutions.II. Ifc=a < n, the equations are inconsistentIII. Ifc=a =n, equation may be inconsistent.Correct answer is '2'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about If the coefficient matrix isCn×n and its corresponding rank iscalso the augmented matrix isAn×(n+1) and its corresponding rank isa,then how many statements given below are incorrect?I. Ifc ≠ a, the equations are inconsistent with an infinite number of solutions.II. Ifc=a < n, the equations are inconsistentIII. Ifc=a =n, equation may be inconsistent.Correct answer is '2'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the coefficient matrix isCn×n and its corresponding rank iscalso the augmented matrix isAn×(n+1) and its corresponding rank isa,then how many statements given below are incorrect?I. Ifc ≠ a, the equations are inconsistent with an infinite number of solutions.II. Ifc=a < n, the equations are inconsistentIII. Ifc=a =n, equation may be inconsistent.Correct answer is '2'. Can you explain this answer?.
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