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For a given system of equations if |A|=0 and (adj A)B≠O(zero matrix), then which of the following is correct regarding the solutions of the given equations?
- a)there will be exactly two solutions
- b)there will be exactly one solution
- c)the solution does not exist
- d)there are one or more solutions
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
For a given system of equations if |A|=0 and (adj A)B≠O(zero matrix...
=0, then the system of equations has either no solutions or infinitely many solutions.
Explanation:
If |A|=0, it means that the determinant of the coefficient matrix A is equal to zero. This implies that the matrix A is singular and its rows or columns are linearly dependent. In other words, there is at least one row (or column) in A that can be expressed as a linear combination of the other rows (or columns).
Now, if we multiply both sides of the given system of equations by the adjugate matrix (adj A), we get:
(adj A)A x = (adj A)B
But since |A|=0, we know that the inverse of A does not exist. However, the adjugate matrix can be used to solve the system of equations in this case. Multiplying both sides of the above equation by (1/|A|), we get:
x = (adj A)B
But since |A|=0, we know that (adj A) is not invertible. This means that there are two possibilities: either (adj A)B=0 or (adj A)B≠0.
If (adj A)B=0, then the system of equations has infinitely many solutions because any vector that is orthogonal to (adj A) (i.e. any vector in the null space of (adj A)) is a solution to the system.
If (adj A)B≠0, then the system of equations has no solutions because the vector (adj A)B is not in the column space of A, which means that there is no x that satisfies the equation Ax=B.
Therefore, if |A|=0 and (adj A)B=0, the system of equations has infinitely many solutions, and if |A|=0 and (adj A)B≠0, the system of equations has no solutions.
Explanation:
If |A|=0, it means that the determinant of the coefficient matrix A is equal to zero. This implies that the matrix A is singular and its rows or columns are linearly dependent. In other words, there is at least one row (or column) in A that can be expressed as a linear combination of the other rows (or columns).
Now, if we multiply both sides of the given system of equations by the adjugate matrix (adj A), we get:
(adj A)A x = (adj A)B
But since |A|=0, we know that the inverse of A does not exist. However, the adjugate matrix can be used to solve the system of equations in this case. Multiplying both sides of the above equation by (1/|A|), we get:
x = (adj A)B
But since |A|=0, we know that (adj A) is not invertible. This means that there are two possibilities: either (adj A)B=0 or (adj A)B≠0.
If (adj A)B=0, then the system of equations has infinitely many solutions because any vector that is orthogonal to (adj A) (i.e. any vector in the null space of (adj A)) is a solution to the system.
If (adj A)B≠0, then the system of equations has no solutions because the vector (adj A)B is not in the column space of A, which means that there is no x that satisfies the equation Ax=B.
Therefore, if |A|=0 and (adj A)B=0, the system of equations has infinitely many solutions, and if |A|=0 and (adj A)B≠0, the system of equations has no solutions.
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Community Answer
For a given system of equations if |A|=0 and (adj A)B≠O(zero matrix...
If A is a singular matrix, then |A|=0
In this case, if (adj A) B≠O, then solution does not exist and the system of equations is called inconsistent.
In this case, if (adj A) B≠O, then solution does not exist and the system of equations is called inconsistent.
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For a given system of equations if |A|=0 and (adj A)B≠O(zero matrix), then which of the following is correct regarding the solutions of the given equations?a)there will be exactly two solutionsb)there will be exactly one solutionc)the solution does not existd)there are one or more solutionsCorrect answer is option 'C'. Can you explain this answer?
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For a given system of equations if |A|=0 and (adj A)B≠O(zero matrix), then which of the following is correct regarding the solutions of the given equations?a)there will be exactly two solutionsb)there will be exactly one solutionc)the solution does not existd)there are one or more solutionsCorrect answer is option 'C'. Can you explain this answer? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about For a given system of equations if |A|=0 and (adj A)B≠O(zero matrix), then which of the following is correct regarding the solutions of the given equations?a)there will be exactly two solutionsb)there will be exactly one solutionc)the solution does not existd)there are one or more solutionsCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For a given system of equations if |A|=0 and (adj A)B≠O(zero matrix), then which of the following is correct regarding the solutions of the given equations?a)there will be exactly two solutionsb)there will be exactly one solutionc)the solution does not existd)there are one or more solutionsCorrect answer is option 'C'. Can you explain this answer?.
For a given system of equations if |A|=0 and (adj A)B≠O(zero matrix), then which of the following is correct regarding the solutions of the given equations?a)there will be exactly two solutionsb)there will be exactly one solutionc)the solution does not existd)there are one or more solutionsCorrect answer is option 'C'. Can you explain this answer? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about For a given system of equations if |A|=0 and (adj A)B≠O(zero matrix), then which of the following is correct regarding the solutions of the given equations?a)there will be exactly two solutionsb)there will be exactly one solutionc)the solution does not existd)there are one or more solutionsCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For a given system of equations if |A|=0 and (adj A)B≠O(zero matrix), then which of the following is correct regarding the solutions of the given equations?a)there will be exactly two solutionsb)there will be exactly one solutionc)the solution does not existd)there are one or more solutionsCorrect answer is option 'C'. Can you explain this answer?.
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