Question
The matrix $\begin{bmatrix}0&\text{amp; }1\\1&\text{amp; }0\end{bmatrix}$ is the matrix reflection in the line:
- x = 1
- x + y = 1
- y = 1
- x = y
✓
Answer
- x = y
Solution:
We know that the reflection matrix through a line $\text{y}=\text{mx}$ making an $\angle \theta$ with x - axis is given as.
$\begin{bmatrix} \cos { 2\theta }&\text{amp;}\sin { 2\theta } \\ \sin { 2\theta } &\text{amp;} -\cos { 2\theta }\end{bmatrix}$
Given transformation matrix is $\begin{bmatrix}0&\text{amp; }1\\1&\text{amp; }0\end{bmatrix}$
$\Rightarrow\cos2\theta=0\sin2\theta=1$
$\Rightarrow 2\theta ={90}^{0}$
$\Rightarrow \theta={45}^{0}$
$\Rightarrow \tan \theta=1$
Hence, the line of reflection is $\text{y}=\text{x}$
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