Let $\mathbf{A}$ be a square matrix all of whose entries are integers. Then which one of the following is…
Let $\mathbf{A}$ be a square matrix all of whose entries are integers. Then which one of the following is true?
If $\operatorname{det} A=\pm 1$, then $A^{-1}$ exists but all its entries are not necessarily integers
If $\operatorname{det} A \neq \pm 1$, then $A^{-1}$ exists and all its entries are non-integers
If $\operatorname{det} A=\pm 1$, then $A^{-1}$ exists and all its entries are integers
If $\operatorname{det} A=\pm 1$, 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 $\operatorname{det} A=\pm 1$ and $A^{-1}=\frac{1}{\operatorname{det}(A)}(\operatorname{adj} A)$
$\Rightarrow$ all entries in $\mathrm{A}^{-1}$ are integers.