Question

行列式的证明|ax+by ay+bz az+bx| |x y z||ay+bz az+bx ax+by| |y z x|

Answer 1

左边=|ax ay az|+|ax ay bx|+|ax bz az|+|ax bz bx|+|by ay az|+D6+D7+|by bz bx|

ay az ax ay az by ay bx ax ay bx by bz az ax bz bx by

az ax ay az ax bz az by ay az by bz bx ax ay bx by bz

=|ax ay az|+ |by bz bx| 【其它有a、b混杂的行列式,因为有两列成比例,值为0】

ay az ax bz bx by

az ax ay bx by bz

=a^3*|x y z|+b^3|y z x| 【提出各列公因子】

y z x z x y

z x y x y z

=a^3*|x y z|-b^3|y x z| 【b行列式二列三列交换】

y z x z y x

z x y x z y

=a^3*|x y z| + b^3*|x y z| 【1、2列再交换】

y z x y z x

z x y z x y

=(a^3+b^3)*|x y z|

y z x

z x y

=右边