Let be the complex number 2 3 +i 2 3 . Then the number of distinct complex number z satisfying | array ccc z+1 & & ^2 & z+ ^2 & 1 ^2 & 1 & z+ array |=0 is equal to

Step-by-step solution

Let A= [ array ccc 1 & & ^2 & ^2 & 1 ^2 & 1 & array ] Now, A^2= [ array lll 0 & 0 & 0 0 & 0 & 0 0 & 0 & 0 array ] and Tr (A)=0,|A|=0 A^3=0 | array ccc z+1 & & ^2 & z+ ^2 & 1 ^2 & 1 & z+ array |=|A+z I|=0 z^3=0 z=0 , the number of z satisfying the given equation is 1 .