" (ii) "|[a,b,c],[a+2x,b+2y,c+2z],[x,y,z]|

Show without expanding at any stage that: det [ [a,b,c],[a+2x,b+2y,c+2z],[x,y,z]] =0

prove that abs[[a,b,c],[a+2x,b+2y,c+2z],[x,y,z]]=0

Without expanding , show that |(a,b,c),(a+2x,b+2y,c+2z),(x,y,z)|= 0 .

Using the property of determinants and without expanding prove the following |{:(a,b,c),(a+2x,b+2y,c+2z),(x,y,z):}|=0

Show that |{:(a,b,c),(a+2x,b+2y,c+2z),(x,y,z):}|=0

Without expanding at any stage, find the value of : |{:(,a,b,c),(,a+2x, b+2y, c+2z),(,x,y,z):}|

Show that abs((a,b,c),(a+2x,b+2y,c+2z),(x,y,z)) =0

Without expanding, show that the value of each of the following determinants is zero: |1^2 2^2 3^2 4^2\ \ \ 2^2 3^2 4^2 5^2\ \ \ 3^2 4^2 5^2 6^2\ \ \ 4^2 5^2 6^2 7^2| (ii) |a b c a+2x b+2y c+2z x y z| (iii) |(2^x+2^(-x))^2(2^x-2^(-x))^2 1(3^x+3^(-x))^2(3^x-3^(-x))^2 1(4^x+4^(-x))^2(4^x-4^(-x))^2 1|

By using properties of determinants. Show that: (i) |[a-b-c,2a,2a],[2b,b-c-a,2b],[2c,2c,c-a-b]|=(a+b+c)^3 (ii) |[x+y+2z, x, y],[ z, y+z+2x, y],[ z, x, z+x+2y]|=2(x+y+z)^3

Prove that |[a x-b y-c z, a y+b x, c x+a z], [a y+b x, b y-c z-a x, b z+c y],[c x+a z, b z+c y, c z-a x-b y]|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot