`" if " x_(i) =a_(i) b_(i) C_(i) i= 1,2,3` are three- digit positive integer such that each `x_(i)` is a mulptiple of 19 then prove that `{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(c_(1),,c_(2),,c_(3)):}` is divisible by 19.

" if " x_(i) =a_(i) b_(i) C_(i), i= 1,2,3 are three- digit positive integer such that each x_(i) is a mulptiple of 19 then prove that det {{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(c_(1),,c_(2),,c_(3)):}} is divisible by 19.

if a_(r) = (cos 2r pi + I sin 2 r pi)^(1//9) then prove that |{:(a_(1),,a_(2),,a_(3)),(a^(4) ,,a^(5),,a_(6)),( a_(7),, a_(8),,a_(9)):}|=0

If a_(i) , i=1,2,…..,9 are perfect odd squares, then |{:(a_(1),a_(2),a_(3)),(a_(4),a_(5),a_(6)),(a_(7),a_(8),a_(9)):}| is always a multiple of

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:

Prove that the value of each the following determinants zero: (a)|{:(a_(1),,la_(1)+mb_(1),,b_(1)),(a_(2),,la_(2)+mb_(2),,b_(2)),( a_(3),,la_(3)+mb_(3),,b_(3)):}|

Let overset(to)(a) =a_(1) hat(i) + a_(2) hat(j) + a_(3) hat(k) , overset(to)(b) = b_(1) hat(i) +b_(2) hat(j) +b_(3) hat(k) " and " overset(to)(c) = c_(1) hat(i) +c_(2) hat(j) + c_(3) hat(k) be three non- zero vectors such that overset(to)(c ) is a unit vectors perpendicular to both the vectors overset(to)(c ) and overset(to)(b) . If the angle between overset(to)(a) " and " overset(to)(n) is (pi)/(6) then |{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(c_(1),,c_(2),,c_(3)):}|^2 is equal to

Let veca=a_(1)hati+a_(2)hatj+a_(3)hatk,vecb=b_(1)hati + b_(2)hatj + b_(3)hatk and vecc=c_(1)hati +c_(2)hatj +c_(3)hatk be three non-zero vectors such that vecc is a unit vector perpendicular to both vectors, veca and vecb . If the angle between veca and vecb is pi//6 "then" |{:(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3)):}|^(2) is equal to

if A_(1) ,B_(1),C_(1) ……. are respectively the cofactors of the elements a_(1) ,b_(1),c_(1)…… of the determinant Delta = |{:(a_(1),,b_(1),,c_(1)),(a_(2),,b_(2),,c_(2)),(a_(3),,b_(3),,c_(3)):}|, Delta ne 0 then the value of |{:(B_(2),,C_(2)),(B_(3),,C_(3)):}| is equal to

Let A=[a_("ij")]_(3xx3) be a matrix such that A A^(T)=4I and a_("ij")+2c_("ij")=0 , where C_("ij") is the cofactor of a_("ij") and I is the unit matrix of order 3. |(a_(11)+4,a_(12),a_(13)),(a_(21),a_(22)+4,a_(23)),(a_(31),a_(32),a_(33)+4)|+5 lambda|(a_(11)+1,a_(12),a_(13)),(a_(21),a_(22)+1,a_(23)),(a_(31),a_(32),a_(33)+1)|=0 then the value of lambda is