The question in your image involves matrix transformations, particularly reflections and rotations in a plane.

• The matrix $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ represents a reflection in the x-axis.
• The matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ represents a reflection in the line $y = x$.

Now, you are asked to find the equivalent transformation after first reflecting in the x-axis and then reflecting in the line $y = x$.

Step-by-Step Solution:

1. First Reflection (in the x-axis):
   The reflection matrix for the x-axis is:
   $R_x = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

2. Second Reflection (in the line $y = x$):
   The reflection matrix for the line $y = x$ is:
   $R_{y=x} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

3. Combined Transformation: To find the resulting transformation, multiply the two matrices in sequence: $R_{total} = R_{y=x} \cdot R_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

   Performing the matrix multiplication: $R_{total} = \begin{pmatrix} 0 \cdot 1 + 1 \cdot 0 & 0 \cdot 0 + 1 \cdot (-1) \\ 1 \cdot 1 + 0 \cdot 0 & 1 \cdot 0 + 0 \cdot (-1) \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$

4. Interpreting the Result: The matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ represents a rotation by $\frac{\pi}{2}$ radians (90 degrees) counterclockwise.

Thus, the combined effect of reflecting first in the x-axis and then in the line $y = x$ is equivalent to rotating counterclockwise by $\frac{\pi}{2}$.

Correct Answer:

• Option A: Rotating anticlockwise by $\frac{\pi}{2}$

Would you like a further explanation, or do you have any additional questions?

Related Questions:

1. What is the transformation matrix for reflection in the y-axis?
2. How do you find the inverse of a transformation matrix?
3. What is the difference between a reflection and a rotation matrix?
4. Can you describe a reflection across a line that isn't the x- or y-axis, such as $y = mx$?
5. How do matrix transformations relate to eigenvalues and eigenvectors?

Tip:

When combining transformations, matrix multiplication must be performed in the correct sequence, as matrix multiplication is not commutative (i.e., order matters).