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Question

Let A be a square matrix all of whose entries are integers, then which of the following is true?

A
If |A|±1, then A1 exist & all its entries are non-integer
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B
If |A|=±1, then A1 exist & all its entries are integer
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C
If |A|=±1,then A1 need not exist
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D
If |A|=±1, then A1 exist but all its entries are not necessarily integers.
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Solution

The correct option is B If |A|=±1, then A1 exist & all its entries are integer
Given A is a square matrix with all entries as integer
(a) Now |A|1,1 mean it may be zero also |A|=0,then A1 does'not exist choice (a) is false.
(b) If |A|=1,1 then A1 certainly exist but A is a square matrix with all integral entries so all cofactors are integers .So adj. A matrix has all integral entries .
A1=1|A|(adjA)=±(adjA)
choice (b) is correct.
(c) |A|=1,1 A1 must exist but given A1 does not exist which is false result choice (c) is incorrect
(d) |A|=1,1 it is true that A1 exist but we have to follow that A has all integral entries but choice (d) says that it is not necessarily that entries are integer which mean adj. A, may or may not be with integral entries. Hence choice (d) is false.

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