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Let AX - B be a system of linear equations where A is an m x n matrix and b is a m x 1 column vector and X is an x 1 column vector of unknown. Which of the following is false?
- a)The system has a solution if and only if, both A and the augmented matrix [A, B] have the same rank.
- b)If m < n and B is the zero vector, then the system has infinitely many solutions.
- c)If m = n and B is non-zero vector, then the system has a unique solution.
- d)The system will have only a trivial solution when m - n, B is the zero vector and rank (A) = n.
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Let AX - B be a system of linear equations where A is an m x n matrix ...
Following are the possibilities for a system of linear equations:
(i) If matrix A and augmented matrix [AB] have same rank, then the system has solution otherwise there is no solution.
(ii) If matrix A and augmented matrix [AB] have same rank which is equal to the number of variables, then the system has unique solution and if B is zero vector then the system have only a trivial solution.
(iii) If matrix A and matrix [AB] have same rank which is less than the number of variables, then the system has infinite solution.
Therefore, option (c) is false because if m - n and B is non-zero vector, then it is not necessary that system has a unique solution, because m is the number of equations (quantity) and not the number of linearly independent equations (quality).
(i) If matrix A and augmented matrix [AB] have same rank, then the system has solution otherwise there is no solution.
(ii) If matrix A and augmented matrix [AB] have same rank which is equal to the number of variables, then the system has unique solution and if B is zero vector then the system have only a trivial solution.
(iii) If matrix A and matrix [AB] have same rank which is less than the number of variables, then the system has infinite solution.
Therefore, option (c) is false because if m - n and B is non-zero vector, then it is not necessary that system has a unique solution, because m is the number of equations (quantity) and not the number of linearly independent equations (quality).
Most Upvoted Answer
Let AX - B be a system of linear equations where A is an m x n matrix ...
C)If n > m, then the system may have infinitely many solutions.
Explanation:
(a) is true, as it is a basic result in linear algebra that a system of linear equations has a solution if and only if the rank of the coefficient matrix A is equal to the rank of the augmented matrix [A, B].
(b) is true, as it follows from the rank-nullity theorem that the dimension of the nullspace of A (i.e. the set of solutions to AX = 0) is n - rank(A). Since m < n,="" the="" nullspace="" has="" dimension="" at="" least="" n="" -="" m="" /> 0, which means that there exist solutions to AX = 0. Therefore, the system AX - B has no solution for any choice of B.
(c) is false, as it follows from the rank-nullity theorem that the dimension of the nullspace of A is n - rank(A). Since n > m, there are more unknowns than equations, so the rank of A cannot be equal to n. Therefore, the dimension of the nullspace is greater than zero, which means that there exist solutions to AX = 0. However, this does not necessarily mean that the system AX - B has infinitely many solutions, as B may not lie in the range of A. If B does lie in the range of A, then the system has a unique solution, and if it does not, then the system has no solution.
Explanation:
(a) is true, as it is a basic result in linear algebra that a system of linear equations has a solution if and only if the rank of the coefficient matrix A is equal to the rank of the augmented matrix [A, B].
(b) is true, as it follows from the rank-nullity theorem that the dimension of the nullspace of A (i.e. the set of solutions to AX = 0) is n - rank(A). Since m < n,="" the="" nullspace="" has="" dimension="" at="" least="" n="" -="" m="" /> 0, which means that there exist solutions to AX = 0. Therefore, the system AX - B has no solution for any choice of B.
(c) is false, as it follows from the rank-nullity theorem that the dimension of the nullspace of A is n - rank(A). Since n > m, there are more unknowns than equations, so the rank of A cannot be equal to n. Therefore, the dimension of the nullspace is greater than zero, which means that there exist solutions to AX = 0. However, this does not necessarily mean that the system AX - B has infinitely many solutions, as B may not lie in the range of A. If B does lie in the range of A, then the system has a unique solution, and if it does not, then the system has no solution.
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Let AX - B be a system of linear equations where A is an m x n matrix and b is a m x 1 column vector and X is an x 1 column vector of unknown. Which of the following is false?a)The system has a solution if and only if, both A and the augmented matrix [A, B] have the same rank.b)If m < n and B is the zero vector, then the system has infinitely many solutions.c)If m = n and B is non-zero vector, then the system has a unique solution.d)The system will have only a trivial solution when m - n, B is the zero vector and rank (A) = n.Correct answer is option 'C'. Can you explain this answer?
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Let AX - B be a system of linear equations where A is an m x n matrix and b is a m x 1 column vector and X is an x 1 column vector of unknown. Which of the following is false?a)The system has a solution if and only if, both A and the augmented matrix [A, B] have the same rank.b)If m < n and B is the zero vector, then the system has infinitely many solutions.c)If m = n and B is non-zero vector, then the system has a unique solution.d)The system will have only a trivial solution when m - n, B is the zero vector and rank (A) = n.Correct answer is option 'C'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Let AX - B be a system of linear equations where A is an m x n matrix and b is a m x 1 column vector and X is an x 1 column vector of unknown. Which of the following is false?a)The system has a solution if and only if, both A and the augmented matrix [A, B] have the same rank.b)If m < n and B is the zero vector, then the system has infinitely many solutions.c)If m = n and B is non-zero vector, then the system has a unique solution.d)The system will have only a trivial solution when m - n, B is the zero vector and rank (A) = n.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let AX - B be a system of linear equations where A is an m x n matrix and b is a m x 1 column vector and X is an x 1 column vector of unknown. Which of the following is false?a)The system has a solution if and only if, both A and the augmented matrix [A, B] have the same rank.b)If m < n and B is the zero vector, then the system has infinitely many solutions.c)If m = n and B is non-zero vector, then the system has a unique solution.d)The system will have only a trivial solution when m - n, B is the zero vector and rank (A) = n.Correct answer is option 'C'. Can you explain this answer?.
Let AX - B be a system of linear equations where A is an m x n matrix and b is a m x 1 column vector and X is an x 1 column vector of unknown. Which of the following is false?a)The system has a solution if and only if, both A and the augmented matrix [A, B] have the same rank.b)If m < n and B is the zero vector, then the system has infinitely many solutions.c)If m = n and B is non-zero vector, then the system has a unique solution.d)The system will have only a trivial solution when m - n, B is the zero vector and rank (A) = n.Correct answer is option 'C'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Let AX - B be a system of linear equations where A is an m x n matrix and b is a m x 1 column vector and X is an x 1 column vector of unknown. Which of the following is false?a)The system has a solution if and only if, both A and the augmented matrix [A, B] have the same rank.b)If m < n and B is the zero vector, then the system has infinitely many solutions.c)If m = n and B is non-zero vector, then the system has a unique solution.d)The system will have only a trivial solution when m - n, B is the zero vector and rank (A) = n.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let AX - B be a system of linear equations where A is an m x n matrix and b is a m x 1 column vector and X is an x 1 column vector of unknown. Which of the following is false?a)The system has a solution if and only if, both A and the augmented matrix [A, B] have the same rank.b)If m < n and B is the zero vector, then the system has infinitely many solutions.c)If m = n and B is non-zero vector, then the system has a unique solution.d)The system will have only a trivial solution when m - n, B is the zero vector and rank (A) = n.Correct answer is option 'C'. Can you explain this answer?.
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