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Question

Let ω1 be a cube root of unity and S be the set of all non-singular matrices of the form
1abω1cω2ω1
where each of a,b, and c is either ω or ω2 Then the number of distinct matrices in the set S is:

A
2
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B
6
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C
4`
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D
8
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Solution

The correct option is A 2
Since, the given matrix 1abω1cω2ω1 is non-singular.
So, Δ=∣ ∣1abω1cω2ω1∣ ∣0
1(1ωc)a(ωω2c)+b(ω2ω2)0
1ω(a+c)+acω20
a+c1 ......(1)
and ac1 ......(2)
So, a=ω or ω2 and c=ω or ω2
If c=ω2, then Δ=0. So cω2
So, c=ω
So, by equation (1), aω2.
Hence, a=ω.
Since, the determinant value is independent of b . So b can be ω or ω2.
Hence, c=ω,a=ω,b=ω or ω2.
So, number of matrices formed will be 2.

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