Question

If $A=[a_{ij}{]}_{3\times 3}$ such that
$a_{ij}=\{\begin{array}{cc}3& \text{when}i=j\\ \left[\frac{i}{j}\right]& \text{when}j>i\\ \left[\frac{j}{i}\right]& \text{when}i>j\end{array}$
then $\left\{\frac{det\left(adj\right(adjA)}{8}\right\}$ is
(where $\{.\}$ represents fractional part function and [.] represents greatest integer function)

• A. $0$
• B. $\frac{1}{8}$
• C. $\frac{1}{16}$
• D. $\frac{1}{4}$

Answer 1

The correct option is B $\frac{1}{8}$

$A=\left[\begin{array}{ccc}3& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right]\Rightarrow |A|=3^3\Rightarrow det\left(adj\right(adjA\left)\right)=(3^3{)}^4$
$\Rightarrow \left\{\frac{det\left(adj\right(adjA)}{8}\right\}=\left\{\frac{9^6}{8}\right\}=\left\{\frac{{(8+1)}^6}{8}\right\}=\frac{1}{8}$