Question

Determine the standard matrix A that first rotates a point clockwise through – radians and then reflects the point about the line x2 = x1 (or al y = x). [Note: The reflection through y = x transforms (x, y]? to [y, x]?.] A =

Answer 1

The standard matrix A for the specified transformation is obtained by multiplying the rotation matrix R for a -π radian rotation by the reflection matrix M about the line y = x, resulting in the matrix A = [0 -1; -1 0].

Step-by-step explanation:

To determine the standard matrix A that first rotates a point clockwise by the angle π radians and then reflects the point about the line y = x, we apply two transformations. The clockwise rotation by π radians is equivalent to a rotation by -π radians counter-clockwise, which is represented by the following rotation matrix:

R = [
cos(-π) -sin(-π)
sin(-π) cos(-π)
] = [
-1 0
0 -1
]

The reflection about the line y = x is represented by the following matrix:

M = [
0 1
1 0
]

To find the overall transformation, we multiply the rotation matrix by the reflection matrix:

A = MR = [
0 1
1 0
] [
-1 0
0 -1
] = [
0 -1
-1 0
]