If trace (A) 0 and a b c=1 where A= [ array lll a & b & c b & c & a c & a & b array ] and A A^ =1 , then the value of a^3+b^3+c^3 is

Step-by-step solution

a b c=1, trace A=a+b+c 0I= [ array lll a & b & c b & c & a c & a & b array ] [ array lll a & b & c b & c & a c & a & b array ]= [ array lll a^2+b^2+c^2 & a b+b c+c a & a c+a b+b c a b+b c+c a & b^2+c^2+a^2 & b c+c a+a b c a+a b+b c & b c+c a+a b & c^2+a^2+b^2 array ] a^2+b^2+c^2=1, a b+b c+c a=0(a+b+c)^2=a^2+b^2+c^2+2(a b+b c+c a)=1 a+b+c=1 a^3+b^3+c^5=3 a b c+(a+b+c) (a^2+b^2+c^2-a b-b c-c a )=3+1(1-0)=4