If is a complex cube root of unity and A= [ array cc & 0 0 & array ] , then A⁵⁰ is equal to

Step-by-step solution

A= [ array ll & 0 0 & array ] Then, A^2= [ array ll & 0 0 & array ] [ array ll & 0 0 & array ]= [ array cc ^2 & 0 0 & ^2 array ] and A^3=A^2 A aligned A^3 & = [ array cc ^2 & 0 0 & ^2 array ] [ array cc & 0 0 & array ] & = [ array cc ^3 & 0 0 & ^3 array ] aligned A^3= [ array ll 1 & 0 0 & 1 array ] [ . is cube root of unity . ^3=1 ] Now, A⁵⁰=A⁴⁸ A^2 aligned & = (A^3 )¹⁶ A^2=(I)¹⁶ A^2 & =I A^2=A^2 & = [ array cc ^2 & 0 0 & ^2 array ]= [ array ll & 0 0 & array ] A⁵⁰ & = A aligned