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[1,4]&&bil[1,5]&&bil[1,6]&&bil[1,9]&&bil[1,10]&&bil[1,11]&&bil[2,3]&&bil[2,5]&&bil[2,6]&&bil[2,7]&&bil[2,8]&&bil[2,9]&&bil[3,5]&&bil[3,7]&&bil[3,8]&&bil[4,5]&&bil[4,10]&&bil[4,11]&&bil[5,6]&&bil[5,7]&&bil[5,11]&&bil[6,7]&&bil[6,9]&&bil[6,11]&&bil[7,8]&&bil[9,10]&&bil[10,11]&&a1>0&&a11>0&&a12>0]

Solve[bil[1,2]&&bil[1,4]&&bil[1,5]&&bil[1,6]&&bil[1,9]&&bil[1,10]&&bil[1,11]&&bil[2,3]&&bil[2,5]&&bil[2,6]&&bil[2,7]&&bil[2,8]&&bil[2,9]&&bil[3,5]&&bil[3,7]&&bil[3,8]&&bil[4,5]&&bil[4,10]&&bil[4,11]&&bil[5,6]&&bil[5,7]&&bil[5,11]&&bil[6,7]&&bil[6,9]&&bil[6,11]&&bil[7,8]&&bil[9,10]&&bil[10,11]&&a1>0&&a11>0&&a12>0]
matzeros = (({{\[Placeholder], 1, \[Placeholder], 1, 1, 0, \[Placeholder], \[Placeholder], 0, 0, 0}, {1, \[Placeholder], 1, \[Placeholder], 1, 0, 0, 0, 0, \[Placeholder], \[Placeholder]}, {\[Placeholder], 1, \[Placeholder], \[Placeholder], 1, \[Placeholder], 0, 0, \[Placeholder], \[Placeholder], \[Placeholder]}, {1, \[Placeholder], \[Placeholder], \[Placeholder], 1, \[Placeholder], \[Placeholder], \[Placeholder], \[Placeholder], 0, 0}, {1, 1, 1, 1, \[Placeholder], 0, 0, \[Placeholder], \[Placeholder], \[Placeholder], 0}, {0, 0, \[Placeholder], \[Placeholder], 0, \[Placeholder], 1, \[Placeholder], 1, \[Placeholder], 1}, {\[Placeholder], 0, 0, \[Placeholder], 0, 1, \[Placeholder], 1, \[Placeholder], \[Placeholder], \[Placeholder]}, {\[Placeholder], 0, 0, \[Placeholder], \[Placeholder], \[Placeholder], 1, \[Placeholder], \[Placeholder], \[Placeholder], \[Placeholder]}, {0, 0, \[Placeholder], \[Placeholder], \[Placeholder], 1, \[Placeholder], \[Placeholder], \[Placeholder], 1, \[Placeholder]}, {0, \[Placeholder], \[Placeholder], 0, \[Placeholder], \[Placeholder], \[Placeholder], \[Placeholder], 1, \[Placeholder], 1}, {0, \[Placeholder], \[Placeholder], 0, 0, 1, \[Placeholder], \[Placeholder], \[Placeholder], 1, \[Placeholder]}}))
vars =(({{a1, 1, a15}, {1, 1, a16}, {1, a9, 0}, {a2, 1, 0}, {a3, a10, 0}, {a4, a11, a17}, {a5, a12, 1}, {0, a13, 1}, {a6, 0, a18}, {a7, 0, 1}, {a8, a14, 1}}))
MatrixForm[(({{Root[1-5 #1-7 ((#1)^2)+((#1)^3)&,3], 1, Root[2-4 #1+((#1)^3)&,3]}, {1, 1, Root[-8-4 #1+6 ((#1)^2)+((#1)^3)&,3]}, {1, Root[23+19 #1-11 ((#1)^2)+((#1)^3)&,2], 0}, {-1, 1, 0}, {1, -1, 0}, {Root[1+19 #1-13 ((#1)^2)+((#1)^3)&,3], 1, 0}, {Root[23+19 #1-11 ((#1)^2)+((#1)^3)&,2], Root[-31+67 #1-29 ((#1)^2)+((#1)^3)&,2], 0}, {Root[-4+12 #1-8 ((#1)^2)+((#1)^3)&,3], Root[32-8 ((#1)^2)+((#1)^3)&,2], Root[5+3 #1-5 ((#1)^2)+((#1)^3)&,2]}, {Root[-8-12 #1+2 ((#1)^2)+((#1)^3)&,3], Root[32-64 #1+12 ((#1)^2)+((#1)^3)&,3], 1}, {0, Root[-8-12 #1+2 ((#1)^2)+((#1)^3)&,3], 1}, {0, 0, 1}, {Root[-4-8 #1-2 ((#1)^2)+((#1)^3)&,3], 0, Root[5+3 #1-5 ((#1)^2)+((#1)^3)&,2]}, {Root[8-4 #1-10 ((#1)^2)+((#1)^3)&,3], 0, 1}, {Root[-32+48 #1-16 ((#1)^2)+((#1)^3)&,3], Root[32-8 ((#1)^2)+((#1)^3)&,2], 1}, {0, 0, -1}}))]

MatrixForm[(({{a1, 1, a15}, {1, 1, a16}, {1, a9, 0}, {a2, 1, 0}, {a3, a10, 0}, {a4, a11, a17}, {a5, a12, 1}, {0, a13, 1}, {a6, 0, a18}, {a7, 0, 1}, {a8, a14, 1}}))/.{a1->7.637519478750682`,a10->1.8509690495304543`,a11->2.387873132949264`,a12->3.5369040834187953`,a13->2.9623886081840296`,a14->2.387873132949264`,a15->1.6751308705666448`,a16->1.350261741133286`,a17->1.8060634335253696`,a18->1.8060634335253696`,a2->11.312650349317327`,a3->3.387873132949261`,a4->6.156325174658665`,a5->2.9623886081840296`,a6->4.156325174658662`,a7->10.312650349317328`,a8->12.31265034931733`,a9->3.387873132949261`}]

(({{7.637519478750682`, 1, 1.6751308705666448`}, {1, 1, 1.350261741133286`}, {1, 3.387873132949261`, 0}, {11.312650349317327`, 1, 0}, {3.387873132949261`, 1.8509690495304543`, 0}, {6.156325174658665`, 2.387873132949264`, 1.8060634335253696`}, {2.9623886081840296`, 3.5369040834187953`, 1}, {0, 2.9623886081840296`, 1}, {4.156325174658662`, 0, 1.8060634335253696`}, {10.312650349317328`, 0, 1}, {12.31265034931733`, 2.387873132949264`, 1}}))