Let A be a square matrix all of whose entries are integers. Then, which one of the following is true ?
A
lf detA=±1, then A−1 exists and all its entries are integers
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B
lf detA=±1, then A−1 need not exist
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C
lf detA=±1, , then A−1 exists but all its entries are not necessarily integers
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D
lf detA≠±1, then A−1 exists and all its entries are non-integers
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Solution
The correct option is A lf detA=±1, then A−1 exists and all its entries are integers Since A has all integer elements, its cofactors will also be integers. So Adj(A) will have integer elements So if |A|=±1 we have A−1=1|A|Adj(A)=±1×Adj(A) And hence A−1 exists, and has all the elements as integers.