MatricesHard

Question

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

Options

A.If detA = ± 1, then A-1 exists but all its entries are not necessarily integers
B.If detA ≠ ± 1, then A-1 exists and all its entries are non-integers
C.If detA = ± 1, then A-1 exiexists and all its entries are integers
D.If detA = ± 1, then A-1 need not exist

Solution

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 = (adj A)
⇒     all entries in A-1 are integers.

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