|[1,1,1],[a^(2),b^(2),c^(2)],[a^(3),b^(3),c^(3)]|

Show that det[[1,1,1a^(2),b^(2),c^(2)a^(3),b^(3),c^(3)]]=(b-c)(c-a)(a-b)(bc+ca+ab)det[[a^(2),b^(2),c^(2)a^(3),b^(3),c^(3)]]=(b-c)(c-a)(a-b)(bc+ca+ab)det[[a^(2),b^(2),c^(2)a^(3),b^(3),c^(3)]]=(b-c)(c-a)(a-b)(bc+ca+ab)

Show that |[a_(1),b_(1),-c_(1)],[-a_(2),-b_(2),c_(2)],[a_(3),b_(3),-c_(3)]|=|[a_(1),b_(1),c_(1)],[a_(2),b_(2),c_(2)],[a_(3),b_(3),c_(3)]|

If |[a_(1),b_(1),c_(1)],[a_(2),b_(2),c_(2)],[a_(3),b_(3),c_(3)]|=5, then the value of |[b_(2)c_(3)-b_(3)c_(2),a_(3)c_(2)-a_(2)c_(3),a_(2)b_(3)-a_(3)b_(2)],[b_(3)c_(1)-b_(1)c_(3),a_(1)c_(3)-a_(3)c_(1),a_(3)b_(1)-a_(1)b_(3)],[b_(1)c_(2)-b_(2)c_(1),a_(2)c_(1)-a_(1)c_(2),a_(1)b_(2)-a_(2)b_(1)]| is

if Delta=det[[a_(1),b_(1),c_(1)a_(2),b_(2),c_(2)a_(3),b_(3),c_(3)]]

if quad /_=[[a_(1),b_(1),c_(1)a_(2),b_(2),c_(2)a_(3),b_(3),c_(3)]]

Let A= |(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3)c_(3))| then the cofactor of a_(31) is:

If A=|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|andB=|{:(c_(1),c_(2),c_(3)),(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)):}| then

If, in D={:[(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))]:}, the co-factor of a_(r)" is "A_(r), then , c_(1)A_(1)+c_(2)A_(2)+c_(3)A_(3)=

If the determinant of the matrix [(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))] is denoted by D, then the determinant of the matrix [(a_(1)+3b_(1)-4c_(1),b_(1),4c_(1)),(a_(2)+3b_(2)-4c_(2),b_(2),4c_(2)),(a_(3)+3b_(3)-4c_(3),b_(3),4c_(3))] will be -