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For a non-trivial solution | A | is
- a)|A| > 0
- b)|A| < 0
- c)|A| = 0
- d)|A| ≠ 0
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
For a non-trivial solution | A | isa)|A| > 0b)|A| < 0c)|A| = 0d)...
Understanding Non-Trivial Solutions
In linear algebra, a system of equations can have different types of solutions: trivial and non-trivial. The question revolves around identifying the conditions for a non-trivial solution of a homogeneous system of equations.
What is a Non-Trivial Solution?
- A non-trivial solution refers to any solution other than the zero vector. In the context of linear systems, it means that the solution exists with at least one component being non-zero.
Determinant and Its Implication
- The condition for a system of linear equations to have non-trivial solutions is closely linked to the determinant of the coefficient matrix.
- If the determinant of the matrix |A| is equal to zero (|A| = 0), the system has infinitely many solutions, including non-trivial solutions.
Analyzing the Options
- a) |A| > 0: This condition indicates the matrix is invertible, leading only to the trivial solution.
- b) |A| < 0:="" a="" negative="" determinant="" doesn’t="" provide="" meaningful="" insight="" into="" the="" existence="" of="" non-trivial="" solutions="" in="" this="" context.="" -="" />c)="" |a|="0:" this="" is="" the="" correct="" condition.="" a="" determinant="" of="" zero="" implies="" the="" presence="" of="" non-trivial="" solutions.="" it="" means="" the="" system="" is="" dependent,="" allowing="" for="" solutions="" other="" than="" the="" trivial="" one.="" -="" d)="" |a|="" ≠="" 0:="" this="" would="" suggest="" that="" the="" system="" has="" only="" the="" trivial="" solution,="" as="" the="" matrix="" would="" be="" invertible.="">Conclusion
- The correct answer is option c) |A| = 0 because it indicates that the system of equations has non-trivial solutions. This reflects the fundamental theorem of linear algebra regarding the relationship between determinants and the solutions of linear systems.
In linear algebra, a system of equations can have different types of solutions: trivial and non-trivial. The question revolves around identifying the conditions for a non-trivial solution of a homogeneous system of equations.
What is a Non-Trivial Solution?
- A non-trivial solution refers to any solution other than the zero vector. In the context of linear systems, it means that the solution exists with at least one component being non-zero.
Determinant and Its Implication
- The condition for a system of linear equations to have non-trivial solutions is closely linked to the determinant of the coefficient matrix.
- If the determinant of the matrix |A| is equal to zero (|A| = 0), the system has infinitely many solutions, including non-trivial solutions.
Analyzing the Options
- a) |A| > 0: This condition indicates the matrix is invertible, leading only to the trivial solution.
- b) |A| < 0:="" a="" negative="" determinant="" doesn’t="" provide="" meaningful="" insight="" into="" the="" existence="" of="" non-trivial="" solutions="" in="" this="" context.="" -="" />c)="" |a|="0:" this="" is="" the="" correct="" condition.="" a="" determinant="" of="" zero="" implies="" the="" presence="" of="" non-trivial="" solutions.="" it="" means="" the="" system="" is="" dependent,="" allowing="" for="" solutions="" other="" than="" the="" trivial="" one.="" -="" d)="" |a|="" ≠="" 0:="" this="" would="" suggest="" that="" the="" system="" has="" only="" the="" trivial="" solution,="" as="" the="" matrix="" would="" be="" invertible.="">Conclusion
- The correct answer is option c) |A| = 0 because it indicates that the system of equations has non-trivial solutions. This reflects the fundamental theorem of linear algebra regarding the relationship between determinants and the solutions of linear systems.
Community Answer
For a non-trivial solution | A | isa)|A| > 0b)|A| < 0c)|A| = 0d)...
An n×n homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions. i.e. For a non-trivial solution ∣A∣=0.
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For a non-trivial solution | A | isa)|A| > 0b)|A| < 0c)|A| = 0d)|A| ≠ 0Correct answer is option 'C'. Can you explain this answer? for JEE 2026 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about For a non-trivial solution | A | isa)|A| > 0b)|A| < 0c)|A| = 0d)|A| ≠ 0Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For a non-trivial solution | A | isa)|A| > 0b)|A| < 0c)|A| = 0d)|A| ≠ 0Correct answer is option 'C'. Can you explain this answer?.
For a non-trivial solution | A | isa)|A| > 0b)|A| < 0c)|A| = 0d)|A| ≠ 0Correct answer is option 'C'. Can you explain this answer? for JEE 2026 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about For a non-trivial solution | A | isa)|A| > 0b)|A| < 0c)|A| = 0d)|A| ≠ 0Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For a non-trivial solution | A | isa)|A| > 0b)|A| < 0c)|A| = 0d)|A| ≠ 0Correct answer is option 'C'. Can you explain this answer?.
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