Given $A=\left ( a_{ij} \right )_{5\times5}$ which is a $5 \times5$ matrix with entries $$a_{ij}=\frac{1}{n_{i}+n_{j}+1}$$ where $n_i,n_j \in \mathbb{N}$ where $0\leq i,j\leq 5$.Then $A$ is positive definite if.
$n_1> n_2 > n_3 > n_4 > n_5$
The solution i tried - The matrix formed under this condition is $$\begin{pmatrix} \frac{1}{n_1+n_1+1}&\frac{1}{n_1+n_2+1} & \frac{1}{n_1+n_3+1} & \frac{1}{n_1+n_4+1} &\frac{1}{n_1+n_5+1} \\ \frac{1}{n_2+n_1+1} & \frac{1}{n_2+n_2+1} & \frac{1}{n_2+n_3+1} & \frac{1}{n_2+n_4+1} &\frac{1}{n_2+n_5+1} \\ \frac{1}{n_3+n_1+1} & \frac{1}{n_3+n_2+1}& \frac{1}{n_3+n_3+1} &\frac{1}{n_3+n_4+1} &\frac{1}{n_3+n_5+1} \\ \frac{1}{n_4+n_1+1}& \frac{1}{n_4+n_2+1} &\frac{1}{n_4+n_3+1} &\frac{1}{n_4+n_4+1} &\frac{1}{n_4+n_5+1} \\ \frac{1}{n_5+n_1+1} &\frac{1}{n_5+n_2+1} & \frac{1}{n_5+n_3+1} &\frac{1}{n_5+n_4+1} & \frac{1}{n_5+n_5+1} \end{pmatrix}$$
the above matrix is symmetric and all the entries are positive.Also we know that a symmetric matrix $A$ is positive definite iff all its eigen values are positive .The other condition for a matrix to be a positive definite is $$x^{T}Ax >0 \forall x \in \mathbb{C}$$ next i am not geting how to solve this question.
Please help
Thankyou
edited-
