If `x+y+z=0` prove that `|a x b y c z c y a z b x b z c x a y|=x y z|a b cc a bb c a|`

Without expanding, prove that |a b c x y z p q r|=|x y z p q r a b c|=|y b q x a p z c r|

Show that |a b c a+2x b+2y c+2z x y z|=0

The product of all values of t , for which the system of equations (a-t)x+b y+c z=0,b x+(c-t)y+a z=0,c x+a y+(b-t)z=0 has non-trivial solution, is |a-c-b-c b-a-b-a c| (b) |a b c b c a c a b| |a c bb a cc b a| (d) |a a+bb+c bb+cc+a cc+a a+b|

Prove that =|a cc-a a+cc bb-c b+c a-bb-c0a-c x y z1+x+y|=0 implies that a ,b ,c are in A.P. or a ,c ,b are in G.P.

If x ,\ y , z are not all zero and if a x+b y+c z=0; b x+c y+a z=0; c x+a y+b z=0 Prove that x : y : z=1:1:1\ or\ 1:omega:omega^2or\ 1:omega^2:omegadot

Value of [[x+y, z,z ],[x, y+z, x],[y, y, z+x]], where x ,y ,z are nonzero real number, is equal to x y z b. 2x y z c. 3x y z d. 4x y z