# 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.?

Jul 14, 2018

#### Explanation:

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

${A}_{1} = \left(\begin{matrix}0 & 1 \\ 1 & 0\end{matrix}\right)$

The matrix of the rotation anticlockwise by ${90}^{\circ}$ is

${A}_{2} = \left(\begin{matrix}0 & - 1 \\ 1 & 0\end{matrix}\right)$

The combination of matrices ${A}_{1}$ and ${A}_{2}$ is

${A}_{2} \cdot {A}_{1} = \left(\begin{matrix}0 & - 1 \\ 1 & 0\end{matrix}\right) \cdot \left(\begin{matrix}0 & 1 \\ 1 & 0\end{matrix}\right)$

$= \left(\begin{matrix}- 1 & 0 \\ 0 & 1\end{matrix}\right)$

The matrix of the reflection in the y-axis is

${A}_{3} = \left(\begin{matrix}- 1 & 0 \\ 0 & 1\end{matrix}\right)$

Therefore,

${A}_{1} \cdot {A}_{2} = {A}_{3}$

For verification, you can take a vector $\left(\begin{matrix}x \\ y\end{matrix}\right)$ and apply the matrices.