Question

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

• A. If $\left|A\right|\ne \pm 1$, then $A^{-1}$ exist & all its entries are non-integer
• B. If $\left|A\right|=\pm 1$, then $A^{-1}$ exist & all its entries are integer
• C. If $\left|A\right|=\pm 1$,then $A^{-1}$ need not exist
• D. If $\left|A\right|=\pm 1$, then $A^{-1}$ exist but all its entries are not necessarily integers.

Answer 1

The correct option is B If $\left|A\right|=\pm 1$, then $A^{-1}$ exist & all its entries are integer

Given A is a square matrix with all entries as integer
(a) Now $\left|A\right|\ne 1,-1$ mean it may be zero also $\left|A\right|=0,$then $A^{-1}$ does'not exist $\therefore$ choice (a) is false.
(b) If $\left|A\right|=1,-1$ then $A^{-1}$ 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 .
$A^{-1}=\frac{1}{\left|A\right|}\left(adjA\right)=\pm \left(adjA\right)$
$\therefore$ choice (b) is correct.
(c) $\left|A\right|=1,-1$ $\therefore$ $A^{-1}$ must exist but given $A^{-1}$ does not exist which is false result $\therefore$ choice (c) is incorrect
(d) $\left|A\right|=1,-1$ it is true that $A^{-1}$ 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.