Question
Let A be a square matrix all of whose entries are integers. Then which one of the following is true?
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If det A = ± 1, then A–1 exists but all its entries are not necessarily integers
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If detA ≠ ± 1, then A–1 exists and all its entries are non-integers
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If detA = ± 1, then A–1 exists and all its entries are integers
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If detA = ± 1, then A–1 need not exist
Solution
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
