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

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