Question

Let $D_k$ be the $k\times k$ matrix with $0$ 's in the main diagonal, unity as the element of $1^{\text{st }}$ row and $(f(k){)}^{\text{th }}$ column and $k$ for all other entries. If $f(x)=x-\{x\}$ where $\{x\}$ denotes the fractional part function then the value of det. $\left(D_2\right)+$ det. $\left(D_3\right)$ equals

Answer 1

Answer: 34

det. $\left(D_2\right)=\left|\begin{array}{cc}0& 1\\ 2& 0\end{array}\right|$
$(f(2)=2\Rightarrow 1^{\text{st }}$ and $2^{\text{nd }}$ column $=1)$
Similarly det. $\left(D_3\right)=\left|\begin{array}{ccc}0& 3& 1\\ 3& 0& 3\\ 3& 3& 0\end{array}\right|$
$\left(f(3)=3\Rightarrow a_{13}=1\right)$
det. $\left(D_2\right)=-2;$ det. $\left(D_3\right)=36$
$\Rightarrow$ det. $\left(D_2\right)+$ det. $\left(D_3\right)=34$