An anticlockwise rotation matrix and reflection in the

x

-axis are

R = (cos θ sin θ - sin θ cos θ), M = (10 0 - 1) .

The columns of

R

are the images of the standard coordinate vectors, while

M (x, y)^{T} = (x, - y)^{T}

ii

Solution

MRM = (cos θ - sin θ sin θ cos θ) = R^{- 1} .

Hence

MRM = R^{a} with a = - 1 .

This is the defining conjugation relation between a rotation and reflection in a dihedral group.

No. Every power

R^{a}

1

, whereas

det (M R^{b}) = det M det (R^{b}) = - 1.

Matrices with different determinants cannot be equal.

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