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

Please see the explanation below

Explanation:

The matrix of the reflection in the line `y=x` is

`A_1=((0,1),(1,0))`

The matrix of the rotation anticlockwise by `90^@` is

`A_2=((0,-1),(1,0))`

The combination of matrices `A_1` and `A_2` is

`A_2*A_1=((0,-1),(1,0))*((0,1),(1,0))`

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

The matrix of the reflection in the y-axis is

`A_3=((-1,0),(0,1))`

Therefore,

`A_1*A_2=A_3`

For verification, you can take a vector `((x),(y))` and apply the matrices.