Mathematics Exam > Mathematics Questions > Which of the following is/are true ?a)Let Ax ...
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Which of the following is/are true ?
- a)Let Ax = b be a system of m linear equations in n unknowns. If this system is consistent, then the dimension of the solution set of Ax = b is n-r, where r = rank of matrix A.
- b)Any system of n linear equations in n unknowns has atmost one solution
- c)Any system of n linear equations in n unknowns has atleast one solution.
- d)None ofthe above
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
Which of the following is/are true ?a)Let Ax = b be a system of m line...
Explanation:
Consistency and Dimension of Solution Set:
- If a system of linear equations Ax = b is consistent, it means that there exists at least one solution to the system.
- The dimension of the solution set of Ax = b is n-r, where r is the rank of matrix A. This is because the dimension of the solution set is equal to the number of free variables, which is n-r.
One Solution for System of Equations:
- Any system of n linear equations in n unknowns can have zero, one, or infinitely many solutions. It is not necessary that there is only one solution.
- For example, consider the system x + y = 2 and 2x + 2y = 4. This system has infinitely many solutions.
Atleast One Solution for System of Equations:
- It is possible for a system of n linear equations in n unknowns to have no solution. This can happen when the equations are inconsistent or when the rank of the coefficient matrix is less than the rank of the augmented matrix.
- Hence, the statement that any system of n linear equations in n unknowns has at least one solution is not always true.
Therefore, the correct answer is option 'D' as none of the given statements are universally true for all systems of linear equations.
Consistency and Dimension of Solution Set:
- If a system of linear equations Ax = b is consistent, it means that there exists at least one solution to the system.
- The dimension of the solution set of Ax = b is n-r, where r is the rank of matrix A. This is because the dimension of the solution set is equal to the number of free variables, which is n-r.
One Solution for System of Equations:
- Any system of n linear equations in n unknowns can have zero, one, or infinitely many solutions. It is not necessary that there is only one solution.
- For example, consider the system x + y = 2 and 2x + 2y = 4. This system has infinitely many solutions.
Atleast One Solution for System of Equations:
- It is possible for a system of n linear equations in n unknowns to have no solution. This can happen when the equations are inconsistent or when the rank of the coefficient matrix is less than the rank of the augmented matrix.
- Hence, the statement that any system of n linear equations in n unknowns has at least one solution is not always true.
Therefore, the correct answer is option 'D' as none of the given statements are universally true for all systems of linear equations.
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Community Answer
Which of the following is/are true ?a)Let Ax = b be a system of m line...
For option (a)
We know that if the system Ax = b of m linear equation in n unknowns is consistent then the solution set Ax = b has n-r+1 linearly independent solution, where r is the rank of matrix A.
Thus the dimension of the solution set of Ax = b is n-r+1.
⇒ option (a) is not ture.
For option (b)
Let 
The system Ax = b is consistent.
We know that if the system 
is consistent and rank (A) < n, then system has infinite solutions.
is consistent and rank (A) < n, then system has infinite solutions.⇒ option (b) is not true
For Option (c)
Clearly Rank (A) ≠ Rank ([A : b])
⇒ This system has no solution.
⇒ Option (c) is not true.
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Which of the following is/are true ?a)Let Ax = b be a system of m linear equations in n unknowns. If this system is consistent, then the dimension of the solution set of Ax = b is n-r, where r = rank of matrix A.b)Any system of n linear equations in n unknowns has atmost one solutionc)Any system of n linear equations in n unknowns has atleast one solution.d)None ofthe aboveCorrect answer is option 'D'. Can you explain this answer?
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Which of the following is/are true ?a)Let Ax = b be a system of m linear equations in n unknowns. If this system is consistent, then the dimension of the solution set of Ax = b is n-r, where r = rank of matrix A.b)Any system of n linear equations in n unknowns has atmost one solutionc)Any system of n linear equations in n unknowns has atleast one solution.d)None ofthe aboveCorrect answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Which of the following is/are true ?a)Let Ax = b be a system of m linear equations in n unknowns. If this system is consistent, then the dimension of the solution set of Ax = b is n-r, where r = rank of matrix A.b)Any system of n linear equations in n unknowns has atmost one solutionc)Any system of n linear equations in n unknowns has atleast one solution.d)None ofthe aboveCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following is/are true ?a)Let Ax = b be a system of m linear equations in n unknowns. If this system is consistent, then the dimension of the solution set of Ax = b is n-r, where r = rank of matrix A.b)Any system of n linear equations in n unknowns has atmost one solutionc)Any system of n linear equations in n unknowns has atleast one solution.d)None ofthe aboveCorrect answer is option 'D'. Can you explain this answer?.
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