The transformation matrix for mirroring a point in x – y plane about the line y = x is given by

The correct answer is the transformation matrix for mirroring a point in the x – y plane about the line y = x, which is given by:

$$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$$

Step-by-Step Explanation:

1. Let us consider any point P(x, y) in the 2D Cartesian coordinate system.

2. When we mirror (or reflect) this point about the line $y=x$, the coordinates of the point swap their positions. That is, the new x-coordinate becomes the original y-coordinate, and the new y-coordinate becomes the original x-coordinate.

3. Mathematically, the coordinates of the reflected point P'(x', y') are related to the original coordinates by the following linear equations:
$$x^{\prime }=y$$
$$y^{\prime }=x$$

4. Expressing these equations as a system of linear equations in terms of both variables x and y:
$$x^{\prime }=0·x+1·y$$
$$y^{\prime }=1·x+0·y$$

5. Writing this system in matrix multiplication form:
$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$$

6. Consequently, the transformation matrix representing reflection about the line y = x is:
$$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$$