Let `omega != 1` be cube root of unity and `S` be the set of all non-singular matrices of the form `[(1,a,b),(omega,1,c),(omega^2,omega,1)],` where each of `a,b, and c` is either `omega` or `omega^2.` Then the number of distinct matrices in the set `S` is (a) `2` (b) `6` (c) `4` (d) `8`

Options

A. 2
B. 6
C. 4
D. 8

Answer

A

Explanation

For being nonsingular,
`|(1,a,b),(omega,1,c),(omega^(2),omega,1)| ne 0`
or `ac omega^(2)-(a+c) omega+1 ne 0`
Hence, number of possible triplets of (a, b, c) is 2.
i.e., `(omega, omega^(2), omega)` and `(omega, omega, omega)`.