Question

Show by using matrix method that a reflection about the line `y=x` followed by rotation about origin through 90° +ve is equivalent to reflection about y-axis.?

Answer 1

See below

Explanation:

Reflection about the line `y = x`

The effect of this reflection is to switch the x and y values of the reflected point. The matrix is:

• `A = ((0,1),(1,0))`

CCW rotation of a point

For CCW rotations about origin by angle `alpha`:

• `R(alpha) = ((cos alpha, - sin alpha),(sin alpha , cos alpha))`

If we combine these in the order suggested:

`bb x' =A \ R(90^o) \ bb x`

`bb x' = ((0,1),(1,0)) ((0, - 1),(1 , 0)) bb x`

`= ((1,0),(0,-1)) bb x`

`implies ((x'),(y')) =((1,0),(0,-1)) ((x),(y)) = ((x),(-y))`

That is equivalent to a reflection in x-axis .

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Making it a CW rotation:

`((x'),(y')) = ((0,1),(1,0)) ((0, 1),(-1 , 0))((x),(y))`

`= ((-1,0),(0,1)) ((x),(y)) = ((-x),(y))`

That is a reflection in the y-axis.