We are given a matrix of the form
$$\left[\begin{array}{cccccccccccccc} 0&0&0&0&0&0&0&0&0&q_{1}&0&1&0&0\\ 0&2p_{2}q_{2}&0&0&0&0&0&0&0&q_{2}&0&0&1&0\\ 0&0&2p_{3}q_{3}&0&0&0&0&0&0&q_{3}&0&0&0&1\\ 0&0&0&2p_{1}q_{1}&0&0&0&0&0&0&q_{1}&1&0&0\\ 0&0&0&0&0&0&0&0&0&0&q_{2}&0&1&0\\ 0&0&0&0&0&2p_{3}q_{3}&0&0&0&0&q_{3}&0&0&1\\ 0&0&0&0&0&0&2p_{1}q_{1}&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&2p_{2}q_{2}&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&1\\ q_{1}&q_{2}&q_{3}&0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&q_{1}&q_{2}&q_{3}&0&0&0&0&0&0&0&0\\ 1&0&0&1&0&0&1&0&0&0&0&0&0&0\\ 0&1&0&0&1&0&0&1&0&0&0&0&0&0\\ 0&0&1&0&0&1&0&0&1&0&0&0&0&0 \end{array}\right]$$
This is an example for $n = 3,$ i.e. it is an $n^2+2n-1$-dimensional matrix,
with all its non-negative entries strictly positive.
We also have that $p_i+q_i = 1,$ and $0 < p_i,q_i.$
I entered this matrix to maxima CAS and it successfully finds its determinant. Unfortunately, I could not generalize from there, neither could I get any hypothesis. How to prove this matrix is always non-singular?
EDIT: all the numbers are reals, and the symbols are positive reals.
As to symbolic computations, I have the following maxima-targeted
format:
[0,0,0,0,0,0,0,0,0,q1,0,1,0,0],
[0,2*p2*q2,0,0,0,0,0,0,0,q2,0,0,1,0],
[0,0,2*p3*q3,0,0,0,0,0,0,q3,0,0,0,1],
[0,0,0,2*p1*q1,0,0,0,0,0,0,q1,1,0,0],
[0,0,0,0,0,0,0,0,0,0,q2,0,1,0],
[0,0,0,0,0,2*p3*q3,0,0,0,0,q3,0,0,1],
[0,0,0,0,0,0,2*p1*q1,0,0,0,0,1,0,0],
[0,0,0,0,0,0,0,2*p2*q2,0,0,0,0,1,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,1],
[q1,q2,q3,0,0,0,0,0,0,0,0,0,0,0],
[0,0,0,q1,q2,q3,0,0,0,0,0,0,0,0],
[1,0,0,1,0,0,1,0,0,0,0,0,0,0],
[0,1,0,0,1,0,0,1,0,0,0,0,0,0],
[0,0,1,0,0,1,0,0,1,0,0,0,0,0]
EDIT: For a concrete example of $q_1 = \ldots = q_n = 1/2,$ one would have $$ \left[ \begin{array}{cccccccccccccc} 0&0&0&0&0&0&0&0&0&{{1}\over{2}}&0&1&0&0\\ 0&{{1}\over{2}}&0&0&0&0&0&0&0&{{1}\over{2}}&0&0&1&0\\\ 0&0&{{1}\over{2}}&0&0&0&0&0&0&{{1}\over{2}}&0&0&0&1\\ 0&0&0&{{1}\over{2}}&0&0&0&0&0&0&{{1}\over{2}}&1&0&0\\ 0&0&0&0&0&0&0&0&0&0&{{1}\over{2}}&0&1&0\\ 0&0&0&0&0&{{1}\over{2}}&0&0&0&0&{{1}\over{2}}&0&0&1\\ 0&0&0&0&0&0&{{1}\over{2}}&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&{{1}\over{2}}&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&1\\ {{1}\over{2}}&{{1}\over{2}}&{{1}\over{2}}&0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&{{1}\over{2}}&{{1}\over{2}}&{{1}\over{2}}&0&0&0&0&0&0&0&0\\ 1&0&0&1&0&0&1&0&0&0&0&0&0&0\\ 0&1&0&0&1&0&0&1&0&0&0&0&0&0\\ 0&0&1&0&0&1&0&0&1&0&0&0&0&0\\ \end{array} \right] $$
EDIT: The Wikipedia page says the matrix is positive definite if and only if all its eigenvalues are positive.
I ran maxima's eigenvalues-function for the above matrix,
and I got the following:
$$\left[ \left[ \left({{-1}\over{2}}-{{\sqrt{3}\,i}\over{2}}\right)\,
\left({{\sqrt{469}\,i}\over{4\,3^{{{3}\over{2}}}}}+{{1}\over{216}}
\right)^{{{1}\over{3}}}+{{37\,\left({{\sqrt{3}\,i}\over{2}}+{{-1
}\over{2}}\right)}\over{36\,\left({{\sqrt{469}\,i}\over{4\,3^{{{3
}\over{2}}}}}+{{1}\over{216}}\right)^{{{1}\over{3}}}}}+{{1}\over{6}}
, \left({{\sqrt{3}\,i}\over{2}}+{{-1}\over{2}}\right)\,\left({{
\sqrt{469}\,i}\over{4\,3^{{{3}\over{2}}}}}+{{1}\over{216}}\right)^{
{{1}\over{3}}}+{{37\,\left({{-1}\over{2}}-{{\sqrt{3}\,i}\over{2}}
\right)}\over{36\,\left({{\sqrt{469}\,i}\over{4\,3^{{{3}\over{2}}}}}
+{{1}\over{216}}\right)^{{{1}\over{3}}}}}+{{1}\over{6}} , \left({{
\sqrt{469}\,i}\over{4\,3^{{{3}\over{2}}}}}+{{1}\over{216}}\right)^{
{{1}\over{3}}}+{{37}\over{36\,\left({{\sqrt{469}\,i}\over{4\,3^{{{3
}\over{2}}}}}+{{1}\over{216}}\right)^{{{1}\over{3}}}}}+{{1}\over{6}}
, {{1}\over{2}} \right] , \left[ 1 , 1 , 1 , 1 \right] \right] $$
Conceptually (if that helps and no-so-obvious from the above representation), this matrix can be seen as a block matrix $$B = \left[\begin{array}{cc}P&V^{T}\\V&\mathbf{0}\end{array}\right]$$