Let A be a $5\times 5$ matrix with entries $a_{ij}=\frac{1}{n_i+n_j+1}$.Then

Question

Let $A$ be a $5\times 5$ matrix with entries $ a_{ij} = \frac{1}{n_i+n_j+1} $,where $ n_i,n_j \in \mathbb{N} $.

Then which of the following cases $A$ is a positive definite matrix ?

1. $ n_i=i $ for all $ i=1,2,3,4,5 $

2. $ n_1< n_2< n_3 < n_4 < n_5 $

3. $ n_1=n_2=n_3=n_4=n_5 $

4. $ n_1> n_2 > n_3 > n_4 > n_5 $

For option 1, I wrote down the matrix , tried to calculate leading principal minors which was a difficult task.

for option 2 and 4 , don't have any clue.

option 3 is trivially false.

Any hint for option 1 , 2 and 4 would be great. THANKS

Answer 1

Note that $$\frac1{m+1}=\int_0^1x^m\,dx.$$ Then for constants $a_1,\ldots,a_5$, if we set $f(x)=a_1x^{n_1}+\cdots+a_5x^{n_5}$ then $$\int_0^1f(x)^2\,dx=\sum_{i,j=1}^5a_ia_j\int_0^1 x^{n_i+n_j}\,dx =\sum_{i,j=1}^n\frac{a_ia_j}{n_i+n_j+1}>0$$ unless $f$ is identically zero on $[0,1]$.