The result of reflecting across the line y = −x and then

Question

rotating 330 degrees counterclockwise around the origin is an
isometry T . Represent T by a 2 × 2 matrix. There is more than
one way to do it. Use the point (1, 1) to check your answer.

Answer 1

The matrix for reflecting across the line y = −x is given by:

[ 0 1 ]
[ 1 0 ]

The matrix for rotation by 330 degrees counterclockwise is equivalent to a rotation of 30 degrees clockwise. The matrix for that rotation is given by:

[ cos(30) sin(30)]
[-sin(30) cos(30)]

Since cos(30) = sqrt(3)/2 and sin(30) = 1/2, the matrix for a rotation of 30 degrees clockwise becomes:

[ sqrt(3)/2 1/2 ]
[-1/2 sqrt(3)/2]

The result of reflecting across the line y = -x and then rotating 330 degrees counterclockwise around the origin is obtained by the multiplication of the matrix for rotation by the matrix for reflection. That is,

T = Rotation Matrix * Reflection Matrix,

which gives:

T = [ sqrt(3)/2 1/2 ] * [ 0 1 ] = [ 1/2 sqrt(3)/2 ]
[-1/2 sqrt(3)/2] [ 1 0 ] [ sqrt(3)/2 -1/2 ]

Now, let's check the answer using the point (1, 1).

T * (1, 1) = [ 1/2 sqrt(3)/2 ] * (1, 1) = ( sqrt(2), 0)
[ sqrt(3)/2 -1/2]

I apologize, there actually seems to have been a mistake when calculating T. The matrix T is computed by multiplying every row in the first matrix by every column in the second matrix, so a correct calculation of T would actually result in:

T = [ 1/2 sqrt(3)/2 ]
[ sqrt(3)/2 -1/2 ]

Sorry about the earlier confusion!