If `[(cos\ (2pi)/7,-sin\ (2pi)/7),(sin\ (2pi)/7,cos\ (2pi)/7)]^k=[(1,0),(0,1)],` then the least positive integral value of `k`, is

Explanation

Given: `[(cos\ (2pi)/7,-sin\ (2pi)/7),(sin\ (2pi)/7,cos\ (2pi)/7)]^k=[(1,0),(0,1)],`

We have to find value of least positive integral `k`

Take `A =[(cos\ (2pi)/7,-sin\ (2pi)/7),(sin\ (2pi)/7,cos\ (2pi)/7)]`

Then

`A^2=A`.`A=[(cos\ (2pi)/7,-sin\ (2pi)/7),(sin\ (2pi)/7,cos\ (2pi)/7)][(cos\ (2pi)/7,-sin\ (2pi)/7),(sin\ (2pi)/7,cos\ (2pi)/7)]`

`A^2=[(cos^2 ((2pi)/7)-sin^2 ((2pi)/7), -2sin((2pi)/7)cos ((2pi)/7)),(2sin ((2pi)/7)cos ((2pi)/7),cos^2 ((2pi)/7)-sin^2 ((2pi)/7))]`

`A^2=[(cos (4pi)/7,-sin\ (4pi)/7),(sin\ (4pi)/7,cos\ (4pi)/7)]`

`A^3=A^2.A=[(cos (4pi)/7,-sin\ (4pi)/7),(sin\ (4pi)/7,cos\ (4pi)/7)][(cos\ (2pi)/7,-sin\ (2pi)/7),(sin\ (2pi)/7,cos\ (2pi)/7)]`

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