MidSol4 - In short, we can write x y A x y B x y . Since...

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In short, we can write µ x y A 7→ µ x 0 y 0 B 7→ µ x 0 y 0 . Since the two lines are perpendicular, we see from the picture that these 3 points are vertices of a right-angled triangle and that the origin is at the midpoint of its hypotenuse. Thus µ x 0 y 0 = µ x y = µ x y . Since the matrix product BA corresponds to the composition of A followed by B ,we conclude µ x y = µ x y , so is the reflection about the origin, i.e. the rotation by 180 . Now we can write the matrix: = µ cos(180 ) sin(180 ) sin(180 ) cos(180 ) = µ 10 0 1 . This can also be seen from µ x y = µ x
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