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

  • If det A = ± 1, then A–1 exists but all its entries are not necessarily integers

  • If detA ≠ ± 1, then A–1 exists and all its entries are non-integers

  • If detA = ± 1, then A–1 exists and all its entries are integers

  • If detA = ± 1, then A–1 need not exist

Answer

C.

If detA = ± 1, then A–1 exists and all its entries are integers

Each entry of A is integer, so the cofactor of every entry is an integer and hence each entry in the adjoint of matrix A is integer. Now detA = ± 1 and A–1 =(1/ det(A)) (adj A)
⇒ all entries in A–1 are integers

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