If the system of equations `x=c y+b z` `y=a z+c x` `z=b x+a y` has a non-trivial solution, show that `a^2+b^2+c^2+2a b c=1`

If the system of linear equations x=cy+bz , y=az+cx , z=bx+ay has a non trivial solution show that a^2+b^2+c^2+2abc=1

If a , b , c are non-zero real numbers and if the system of equations (a-1)x=y=z (b-1)y=z+x (c-1)z=x+y has a non-trivial solution, then prove that a b+b c+c a=a b c

If a , b , c are distinct real numbers and the system of equations a x+a^2y+(a^3+1)z=0 b x+b^2y+(b^3+1)z=0 c x+c^2y+(c^3+1)=0 has a non-trivial solution, show that a b c=-1

If a , b , c are distinct real numbers and the system of equations a x+a^2y+(a^3+1)z=0 b x+b^2y+(b^3+1)z=0 c x+c^2y+(c^3+1)=0 has a non-trivial solution, show that a b c=-1

If a , b , c are non-zero real numbers and if the system of equations (a-1)x=y+z (b-1)y=z+x (c-1)z=x+y has a non-trivial solution, then ab+bc+ca equals to (A) abc (B) a^2 - b^2 + c^2 (C) a+b -c (D) None of these

The system of equations ax + 4y + z = 0,bx + 3y + z = 0, cx + 2y + z = 0 has non-trivial solution if a, b, c are in

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) |[a, -c, -b], [-c, b, -a], [-b, -a, c]| (b) |[a, b, c], [b, c, a], [c, a, b]| (c) |[a, c, b], [b, a, c], [c, b, a]| (d) |[a, a+b, b+c], [b, b+c, c+a], [c, c+a, a+b]|

The system of equations -2x+y+z=a x-2y+z=b x+y-2z=c has

Prove that the system of equations in xa +y+z=0 , x +by +z=0 , x +y+cz=0 has a non - trivial solution then 1/(1-a) + 1/(1-b)+ 1/(1-c)=

Show that the homogenous system of equations x - 2y + z = 0, x + y - z = 0, 3 x + 6y - 5z = 0 has a non-trivial solution. Also find the solution