Q69 / 105 · JEE Main 2008 · Mathematics

Let $\mathbf{A}$ be a square matrix all of whose entries are integers. Then which one of the following is true?

  • A
    If $\operatorname{det} A=\pm 1$, then $A^{-1}$ exists but all its entries are not necessarily integers
  • B
    If $\operatorname{det} A \neq \pm 1$, then $A^{-1}$ exists and all its entries are non-integers
  • Correct answer
    C
    If $\operatorname{det} A=\pm 1$, then $A^{-1}$ exists and all its entries are integers
  • D
    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.

Hence, the correct answer is option: 'C'.

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