求行列式
求行列式
\[
D=\begin{vmatrix}
\lambda-1 & -1 & -1 &\cdots & -1\\
-1 & \lambda-1 & -1 &\cdots & -1\\
-1 & -1 &\lambda-1 &\cdots & -1\\
\vdots &\vdots &\vdots &\ddots & \vdots\\
-1 & -1 & -1 & \cdots &\lambda-1
\end{vmatrix}_{n}
\]
设 $c_i\neq 0, i=1,2,\ldots,n$. 求行列式
\[
D=\begin{vmatrix}
\lambda-1 & -\frac{c_1}{c_2} & -\frac{c_1}{c_3} &\cdots & -\frac{c_1}{c_n}\\
-\frac{c_2}{c_1} & \lambda-1 & -\frac{c_2}{c_3} &\cdots & -\frac{c_2}{c_n}\\
-\frac{c_3}{c_1} & -\frac{c_3}{c_2} &\lambda-1 &\cdots & -\frac{c_3}{c_n}\\
\vdots &\vdots &\vdots &\ddots & \vdots\\
-\frac{c_n}{c_1} & -\frac{c_n}{c_2} & -\frac{c_n}{c_3} & \cdots &\lambda-1
\end{vmatrix}_{n}
\]
这两个行列式都等于 $\lambda^{n-1}(\lambda-n)$.