Question

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

• A. lf $detA=\pm 1$, then $A^{-1}$ exists and all its entries are integers
• B. lf $detA=\pm 1$, then $A^{-1}$ need not exist
• C. lf $detA=\pm 1$, , then $A^{-1}$ exists but all its entries are not necessarily integers
• D. lf $detA\ne \pm 1$, then $A^{-1}$ exists and all its entries are non-integers

Answer 1

The correct option is A lf $detA=\pm 1$, then $A^{-1}$ exists and all its entries are integers

Since A has all integer elements, its cofactors will also be integers. So $Adj\left(A\right)$ will have integer elements
So if $|A|=\pm 1$
we have $A^{-1}=\frac{1}{|A|}Adj\left(A\right)=\pm 1\times Adj\left(A\right)$
And hence $A^{-1}$ exists, and has all the elements as integers.