Let `omega!=1` be cube root of unity and `S` be the set of all non-singular matrices of the form `[1a bomega1comega^2theta1],w h e r e` each of `a ,b ,a n dc` is either `omegaoromega^2dot` 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 the given matrix to be non singular
`therefore abs((1, a,b),(omega,1,c),(omega^(2),omega,1))ne0`
`rArr 1-(a+c) omega+acomega^(2) ne0`
`rArr (1- a omega) (1-comega) ne 0`
`rArr a ne omega and c ne omega^(2)`
`because` a, b and c are complex cube roots of unity .
`therefore ` a and c can take only and value i.e., `omega` while b and take two
values i. e. , `omega and omega^(2)`
`therefore` Total number of distinct = 2