【本版第200个主帖】两道n阶行列式

青青子衿

真幸运，抢到了本版第200个主帖
“嵌入型”行列式
\[
D_{n,I}=\left|\begin{array}{ccccccc}
a_1&b_1&b_2& \cdots &b_{n-3}&b_{n-2}&b_{n-1}\\
b_1&a_2&b_2& \cdots &b_{n-3}&b_{n-2}&b_{n-1}\\
b_1&b_2&a_3& \cdots &b_{n-3}&b_{n-2}&b_{n-1}\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\
b_1&b_2&b_3& \cdots &a_{n-2}&b_{n-2}&b_{n-1}\\
b_1&b_2&b_3& \cdots &b_{n-2}&a_{n-1}&b_{n-1}\\
b_1&b_2&b_3& \cdots &b_{n-2}&b_{n-1}&a_{n}
\end{array}\right|=\left(\sum_{j=1}^{n-1}b_j\frac{\displaystyle\prod_{i=1}^j(a_i-b_i)}{\displaystyle\prod_{i=1}^j(a_{i+1}-b_i)}+a_1\right)\prod_{i=1}^{n-1}(a_{i+1}-b_i)
\]
“替换型”行列式
\[
D_{n,II}=\left|\begin{array}{ccccccc}
a_1&b_2&b_3& \cdots &b_{n-2}&b_{n-1}&b_{n}\\
b_1&a_2&b_3& \cdots &b_{n-2}&b_{n-1}&b_{n}\\
b_1&b_2&a_3& \cdots &b_{n-2}&b_{n-1}&b_{n}\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\
b_1&b_2&b_3& \cdots &a_{n-2}&b_{n-1}&b_{n}\\
b_1&b_2&b_3& \cdots &b_{n-2}&a_{n-1}&b_{n}\\
b_1&b_2&b_3& \cdots &b_{n-2}&b_{n-1}&a_{n}
\end{array}\right|=\left(\sum_{j=1}^n\frac{b_j}{a_j-b_j}+1\right)\prod_{i=1}^{n}(a_i-b_i)
\]

tommywong

$\displaystyle D_n=
\begin{vmatrix}
a_1 & c_1 & c_2 & \cdots & c_{n-3} & c_{n-2} & c_{n-1}\\
b_1 & a_2 & c_2 & \cdots & c_{n-3} & c_{n-2} & c_{n-1}\\
b_1 & b_2 & a_3 & \cdots & c_{n-3} & c_{n-2} & c_{n-1}\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots\\
b_1 & b_2 & b_3 & \cdots & a_{n-2} & c_{n-2} & c_{n-1}\\
b_1 & b_2 & b_3 & \cdots & b_{n-2} & a_{n-1} & c_{n-1}\\
b_1 & b_2 & b_3 & \cdots & b_{n-2} & b_{n-1} & a_n
\end{vmatrix}
=(a_n-c_{n-1})D_{n-1}+c_{n-1}\prod_{i=1}^{n-1}(a_i-b_i)$

$\displaystyle=(a_1+\sum_{j=1}^{n-1} c_j\prod_{i=1}^j \frac{a_i-b_i}{a_{i+1}-c_i})\prod_{i=1}^{n-1}(a_{i+1}-c_i)$

$c_i=b_{i+1}$

$\displaystyle(a_1+\sum_{j=1}^{n-1} c_j\prod_{i=1}^j \frac{a_i-b_i}{a_{i+1}-c_i})\prod_{i=1}^{n-1}(a_{i+1}-c_i)
=(a_1+\sum_{j=1}^{n-1}b_{j+1}\frac{a_1-b_1}{a_{j+1}-b_{j+1}})\prod_{i=1}^{n-1}(a_{i+1}-b_{i+1})$

$\displaystyle=(1+\sum_{j=0}^{n-1}\frac{b_{j+1}}{a_{j+1}-b_{j+1}})\prod_{i=0}^{n-1}(a_{i+1}-b_{i+1})$