Let 1 be a cube root of unity and S be the set of all non-singular matrices of the form [ array ccc 1 & a & b & 1 & c ^2 & & 1 array ] , where each of a, b and c is either or ^2 . Then, the number of distinct matrices in the set S is

Step-by-step solution

|A| 0 , as non-singular. aligned & | array ccc 1 & a & b & 1 & c ^2 & & 1 array | 0 & 1(1-c )-a ( -c ^2 ) & +b ( ^2- ^2 ) 0 & 1-c -a +a c ^2 0 & aligned aligned & (1-c )(1-a ) 0 & a 1 , c 1 a= , c= aligned and b , ^2 2 solutions