The determinant computing of some special matrices

In the volume computing of some special geometry in $n$ -dimensions space, the determinant values of some special matrices are computed time and again. Because the computing methods for that does not been found, I have no choice to deduce these computing equations by myself. After the tedious deductions, the concise results was got as follows.

1.

$\det\left[\begin{array}{cccc} \frac{1}{1-\lambda_{1}^{2}} & \frac{1}{1-\lambda_{1}\lambda_{2}} & \cdots & \frac{1}{1-\lambda_{1}\lambda_{n}}\\ \frac{1}{1-\lambda_{1}\lambda_{2}} & \frac{1}{1-\lambda_{2}^{2}} & \cdots & \frac{1}{1-\lambda_{2}\lambda_{n}}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{1}{1-\lambda_{1}\lambda_{n}} & \frac{1}{1-\lambda_{2}\lambda_{n}} & \cdots & \frac{1}{1-\lambda_{n}^{2}} \end{array}\right]=\left[\prod_{1\leq j_{1}

2.

$\det\left[\begin{array}{cccc} \frac{1}{2\lambda_{1}} & \frac{1}{\lambda_{1}+\lambda_{2}} & \cdots & \frac{1}{\lambda_{1}+\lambda_{n}}\\ \frac{1}{\lambda_{1}+\lambda_{2}} & \frac{1}{2\lambda_{2}} & \cdots & \frac{1}{\lambda_{2}+\lambda_{n}}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{1}{\lambda_{1}+\lambda_{n}} & \frac{1}{\lambda_{2}+\lambda_{n}} & \cdots & \frac{1}{2\lambda_{n}} \end{array}\right]=\left[\prod_{1\leq j_{1}

3.

$\sum_{(k_{1},k_{2},\cdots,k_{n})\in\Omega_{0,\infty}^{n}}\det\left(\left[\begin{array}{cccc} \lambda_{1}^{k_{1}} & \lambda_{1}^{k_{2}} & \cdots & \lambda_{1}^{k_{n}}\\ \lambda_{2}^{k_{1}} & \lambda_{2}^{k_{2}} & \cdots & \lambda_{2}^{k_{n}}\\ \vdots & \vdots & \ddots & \vdots\\ \lambda_{n}^{k_{1}} & \lambda_{n}^{k_{2}} & \cdots & \lambda_{n}^{k_{n}} \end{array}\right]\right)=\left(\prod_{1\leq j_{1}

where $\Omega_{0,\infty}^{n}$ is constituted by the all possible multi-tuple $(k_{1},k_{2},\cdots,k_{n})$ which elements are picked from the natural number set $\{0,1,2,\cdots\}$ and sorted by the values.