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If every minor of order r of a matrix A is zero, then rank of A is
- a)greater than r
- b)equal to r
- c)less than or equal to r
- d)less than r
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
If every minor of order r of a matrix A is zero, then rank of A isa)gr...
Explanation:
Rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. If every minor of order r of a matrix A is zero, it means that all submatrices of order r formed by selecting any r rows and r columns from matrix A have determinant equal to zero.
Reasoning:
- If every minor of order r of a matrix A is zero, it implies that the rows or columns of the matrix are linearly dependent.
- This means that the rank of the matrix cannot be greater than r because there cannot be more than r linearly independent rows or columns.
- However, the rank of the matrix can be less than r because even though all minors of order r are zero, it is possible that some minors of order less than r are non-zero and contribute to the rank of the matrix.
Conclusion:
Therefore, if every minor of order r of a matrix A is zero, the rank of the matrix is less than r. This is because the presence of zero minors of order r indicates linear dependence among rows or columns, limiting the maximum number of linearly independent rows or columns in the matrix to be less than r.
Rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. If every minor of order r of a matrix A is zero, it means that all submatrices of order r formed by selecting any r rows and r columns from matrix A have determinant equal to zero.
Reasoning:
- If every minor of order r of a matrix A is zero, it implies that the rows or columns of the matrix are linearly dependent.
- This means that the rank of the matrix cannot be greater than r because there cannot be more than r linearly independent rows or columns.
- However, the rank of the matrix can be less than r because even though all minors of order r are zero, it is possible that some minors of order less than r are non-zero and contribute to the rank of the matrix.
Conclusion:
Therefore, if every minor of order r of a matrix A is zero, the rank of the matrix is less than r. This is because the presence of zero minors of order r indicates linear dependence among rows or columns, limiting the maximum number of linearly independent rows or columns in the matrix to be less than r.
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If every minor of order r of a matrix A is zero, then rank of A isa)greater than rb)equal to rc)less than or equal to rd)less than rCorrect answer is option 'D'. Can you explain this answer?
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If every minor of order r of a matrix A is zero, then rank of A isa)greater than rb)equal to rc)less than or equal to rd)less than rCorrect answer is option 'D'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about If every minor of order r of a matrix A is zero, then rank of A isa)greater than rb)equal to rc)less than or equal to rd)less than rCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If every minor of order r of a matrix A is zero, then rank of A isa)greater than rb)equal to rc)less than or equal to rd)less than rCorrect answer is option 'D'. Can you explain this answer?.
If every minor of order r of a matrix A is zero, then rank of A isa)greater than rb)equal to rc)less than or equal to rd)less than rCorrect answer is option 'D'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about If every minor of order r of a matrix A is zero, then rank of A isa)greater than rb)equal to rc)less than or equal to rd)less than rCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If every minor of order r of a matrix A is zero, then rank of A isa)greater than rb)equal to rc)less than or equal to rd)less than rCorrect answer is option 'D'. Can you explain this answer?.
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