biggermat=(({{-1, 1, \[Placeholder], 1, 1, \[Placeholder], \[Placeholder], 0, 0, \[Placeholder], 0, \[Placeholder], \[Placeholder], \[Placeholder]}, {1, -1, 1, 1, \[Placeholder], 1, 1, \[Placeholder], 0, 0, 0, 0, 0, \[Placeholder]}, {\[Placeholder], 1, -1, 1, \[Placeholder], \[Placeholder], 1, \[Placeholder], \[Placeholder], 0, \[Placeholder], 0, \[Placeholder], 0}, {1, 1, 1, -1, 1, \[Placeholder], \[Placeholder], 0, \[Placeholder], \[Placeholder], 0, 0, \[Placeholder], 0}, {1, \[Placeholder], \[Placeholder], 1, -1, 1, \[Placeholder], 0, 0, \[Placeholder], \[Placeholder], \[Placeholder], \[Placeholder], 0}, {\[Placeholder], 1, \[Placeholder], \[Placeholder], 1, -1, 1, \[Placeholder], 0, \[Placeholder], \[Placeholder], \[Placeholder], 0, 0}, {\[Placeholder], 1, 1, \[Placeholder], \[Placeholder], 1, -1, \[Placeholder], \[Placeholder], 0, \[Placeholder], \[Placeholder], 0, 0}, {0, \[Placeholder], \[Placeholder], 0, 0, \[Placeholder], \[Placeholder], -1, 1, \[Placeholder], 1, \[Placeholder], \[Placeholder], 1}, {0, 0, \[Placeholder], \[Placeholder], 0, 0, \[Placeholder], 1, -1, \[Placeholder], 1, \[Placeholder], 1, 1}, {\[Placeholder], 0, 0, \[Placeholder], \[Placeholder], \[Placeholder], 0, \[Placeholder], \[Placeholder], -1, \[Placeholder], 1, 1, 1}, {0, 0, \[Placeholder], 0, \[Placeholder], \[Placeholder], \[Placeholder], 1, 1, \[Placeholder], -1, 1, \[Placeholder], \[Placeholder]}, {\[Placeholder], 0, 0, 0, \[Placeholder], \[Placeholder], \[Placeholder], \[Placeholder], \[Placeholder], 1, 1, -1, \[Placeholder], 1}, {\[Placeholder], 0, \[Placeholder], \[Placeholder], \[Placeholder], 0, 0, \[Placeholder], 1, 1, \[Placeholder], \[Placeholder], -1, 1}, {\[Placeholder], \[Placeholder], 0, 0, 0, 0, 0, 1, 1, 1, \[Placeholder], 1, 1, -1}}))
matzeros=(({{\[Placeholder], 1, 1, \[Placeholder], \[Placeholder], 0, 0, \[Placeholder], 0, \[Placeholder]}, {1, \[Placeholder], \[Placeholder], 1, 1, \[Placeholder], 0, 0, 0, 0}, {1, \[Placeholder], \[Placeholder], 1, \[Placeholder], 0, 0, \[Placeholder], \[Placeholder], \[Placeholder]}, {\[Placeholder], 1, 1, \[Placeholder], 1, \[Placeholder], 0, \[Placeholder], \[Placeholder], 0}, {\[Placeholder], 1, \[Placeholder], 1, \[Placeholder], \[Placeholder], \[Placeholder], 0, \[Placeholder], 0}, {0, \[Placeholder], 0, \[Placeholder], \[Placeholder], \[Placeholder], 1, \[Placeholder], 1, \[Placeholder]}, {0, 0, 0, 0, \[Placeholder], 1, \[Placeholder], \[Placeholder], 1, 1}, {\[Placeholder], 0, \[Placeholder], \[Placeholder], 0, \[Placeholder], \[Placeholder], \[Placeholder], \[Placeholder], 1}, {0, 0, \[Placeholder], \[Placeholder], \[Placeholder], 1, 1, \[Placeholder], \[Placeholder], \[Placeholder]}, {\[Placeholder], 0, \[Placeholder], 0, 0, \[Placeholder], 1, 1, \[Placeholder], \[Placeholder]}}))
vars = (({{a31, 1, a141}, {1, 1, a142}, {a35, 1, 0}, {a36, a46, 0}, {1, a47, 0}, {a38, 0, 1}, {a39, a49, 1}, {0, a410, 1}, {a311, 0, a1411}, {a313, a413, 1}}))

bil[a_,b_]:=1/2((vars[[a,3]]^2+vars[[a,2]]^2-1)*(vars[[b,1]]+vars[[b,2]])/(vars[[a,1]]+vars[[a,2]])+(vars[[b,3]]^2+vars[[b,2]]^2-1)*(vars[[a,1]]+vars[[a,2]])/(vars[[b,1]]+vars[[b,2]]))-vars[[a,3]]*vars[[b,3]]-vars[[a,2]]*vars[[b,2]]==matzeros[[a,b]]

NSolve[bil[1,2]&&bil[1,3]&&bil[1,6]&&bil[1,7]&&bil[1,9]&&bil[2,4]&&bil[2,5]&&bil[2,7]&&bil[2,8]&&bil[2,9]&&bil[2,10]&&bil[3,4]&&bil[3,6]&&bil[3,7]&&bil[4,5]&&bil[4,7]&&bil[4,10]&&bil[5,8]&&bil[5,10]&&bil[6,7]&&bil[6,9]&&bil[7,9]&&bil[7,10]&&bil[8,10]&&a311>0&&a49>0&&a142>0&&a1411>0&&a36>0&&a47>0]

{a141,a1411,a142,a31,a311,a313,a35,a36,a38,a39,a410,a413,a46,a47,a49}/. %23

RootApproximant[{2.4697275952269337`,2.049777197312833`,1.2963318565217516`,10.069648589016966`,4.705241457994818`,3.085629640185284`,6.259316253508969`,3.7605551381951643`,8.964266836885285`,4.259025378890466`,3.085629640185284`,4.259025378890466`,2.810332335507997`,3.7605551381951643`,1.619611817809534`}]