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Let $$A$$ be a square matrix all of whose entries are integers. Then which one of the following is true?

JEE · Math · previous-year question

  1. A.If det $$A = \pm 1,$$ then $${A^{ - 1}}$$ exists but all its entries are not necessarily integers
  2. B.If det $$A \ne \pm 1,$$ then $${A^{ - 1}}$$ exists and all its entries are non integers
  3. C.If det $$A = \pm 1,$$ then $${A^{ - 1}}$$ exists but all its entries are integerscorrect
  4. D.If det $$A = \pm 1,$$ then $${A^{ - 1}}$$ need not exists

Answer

C. If det $$A = \pm 1,$$ then $${A^{ - 1}}$$ exists but all its entries are integers

Explanation

As all entries of square matrix $$A$$ are integers, therefore all co-factors should also be integers. If det $$A = \pm 1\,\,$$ then $${A^{ - 1}}\,\,$$ exists. Also all entries of $${A^{ - 1}}$$ are integers.

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