Check the conjecture at http://mathoverflow.net/questions/204460/how-to-prove-this-determinant-is-positive/204470#204470

check.py

from numpy import  bmat, zeros , array , eye , dot 
from scipy.linalg import det , expm  


B1 = array([[2, 1, 1, 3, 2],
            [1, 2, 1, 3, 1],
            [2, 3, 1, 2, 2],
            [1, 3, 1, 3, 1],
            [2, 3, 3, 1, 3]]
           )

B2 = array([[3, 3, 2, 2, 3],
            [3, 3, 2, 1, 3],
            [2, 3, 2, 1, 3],
            [3, 3, 3, 2, 2],
            [1, 2, 3, 2, 3]]
           )

B3 = array([[3, 2, 3, 1, 2],
            [2, 2, 2, 3, 3],
            [3, 1, 3, 3, 2],
            [3, 3, 3, 2, 3],
            [3, 2, 1, 2, 1]]
           )

A1 = bmat([ [ zeros((5,5), float), B1], [B1.transpose(), zeros((5,5), float) ]])  
A2 = bmat([ [ zeros((5,5), float), B2], [B2.transpose(), zeros((5,5), float) ]])  
A3 = bmat([ [ zeros((5,5), float), B3], [B3.transpose(), zeros((5,5), float) ]])  

print det(eye(10) +  dot( dot(expm(A1), expm(A2)), expm(A3) ) )