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Let A be a square matrix all of whose entries are integers. Then which one of the following is true?
  • a)
    If det A = ± 1, then A–1 exists but all its entries are not necessarily integers
  • b)
    If det A ≠ ± 1, then A–1 exists and all its entries are non integers
  • c)
    If det A = ± 1, then A–1 exists but all its entries are integers
  • d)
    If det A = ± 1, then A–1 need not exists
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Let A be a square matrix all of whose entries are integers. Then which...
∵ All entries of square matrix A are integers, therefore all  cofactors should also be integers.
If det A = ± 1 then A–1 exists. Also all entries of A–1 are integers.
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Most Upvoted Answer
Let A be a square matrix all of whose entries are integers. Then which...
Explanation:

Given:
- A is a square matrix with all integer entries
- det(A) = ±1

Explanation:

If det A = ±1:
- If the determinant of matrix A is ±1, then A is invertible.
- This means that A has an inverse denoted by A^-1.
- Since A is a square matrix with all integer entries, the inverse A^-1 also exists.
- Therefore, if det A = ±1, then A^-1 exists.

But all its entries are not necessarily integers:
- While A^-1 exists when det A = ±1, it does not necessarily mean that all the entries of A^-1 will be integers.
- The entries of A^-1 can be fractions or decimals, even if the entries of A are integers.

Conclusion:
- Option (c) is true: If det A = ±1, then A^-1 exists but all its entries are not necessarily integers.
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Let A be a square matrix all of whose entries are integers. Then which one of the following is true?a)If det A = ± 1, then A–1 exists but all its entries are not necessarily integersb)If det A ≠ ± 1, then A–1 exists and all its entries are non integersc)If det A = ± 1, then A–1 exists but all its entries are integersd)If det A = ± 1, then A–1 need not existsCorrect answer is option 'C'. Can you explain this answer?
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Let A be a square matrix all of whose entries are integers. Then which one of the following is true?a)If det A = ± 1, then A–1 exists but all its entries are not necessarily integersb)If det A ≠ ± 1, then A–1 exists and all its entries are non integersc)If det A = ± 1, then A–1 exists but all its entries are integersd)If det A = ± 1, then A–1 need not existsCorrect answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let A be a square matrix all of whose entries are integers. Then which one of the following is true?a)If det A = ± 1, then A–1 exists but all its entries are not necessarily integersb)If det A ≠ ± 1, then A–1 exists and all its entries are non integersc)If det A = ± 1, then A–1 exists but all its entries are integersd)If det A = ± 1, then A–1 need not existsCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A be a square matrix all of whose entries are integers. Then which one of the following is true?a)If det A = ± 1, then A–1 exists but all its entries are not necessarily integersb)If det A ≠ ± 1, then A–1 exists and all its entries are non integersc)If det A = ± 1, then A–1 exists but all its entries are integersd)If det A = ± 1, then A–1 need not existsCorrect answer is option 'C'. Can you explain this answer?.
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