My problem is on the specific determinant.
det(na1+b1na2+b2na3+b3nb1+c1nb2+c2nb3+c3nc1+a1nc2+a2nc3+a3)=(n+1)(n2−n+1)det(a1a2a3b1b2b3c1c2c3)
All I can do is prove the factor (n+1) and I think that we have to work only on one column and the do the exact same thing to the others.
Answer
Use multilinearity of determinant:
|na1+b1na2+b2na3+b3nb1+c1nb2+c2nb3+c3nc1+a1nc2+a2nc3+a3|=|na1na2na3nb1nb2nb3nc1nc2nc3|+|na1na2b3nb1nb2c3nc1nc2a3|+
+|na1b2na3nb1c2nb3nc1a2nc3|+|na1b2b3nb1c2c3nc1a2a3|+|b1na2na3c1nb2nb3a1nc2nc3|+…
Observe that if we put
Δ=|a1a2a3b1b2b3c1c2c3|
then we have that the four first determinants above equal (factor out constants from rows/columns):
n3Δ+n2Δ+n2|a1b2a3b1c2b3c1a2c3|+n|a1b2b3b1c2c3c1a2a3|+…
Well, develop the other three determinants left and sum up all.
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