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

The correct answer is the matrix shown in image_2.png, which is:
$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

Step-by-Step Derivation:

1. Let a point in the 2D Cartesian coordinate system be represented by coordinates $(x,y)$.

2. We want to find the coordinates of the mirrored point $(x^{\prime },y^{\prime })$ after reflection about the line $y=x$.

3. Under reflection across the line $y=x$, the coordinates of the point are swapped. That is, the new x-coordinate becomes the original y-coordinate, and the new y-coordinate becomes the original x-coordinate:
$x^{\prime }=y$
$y^{\prime }=x$

4. We can rewrite these equations to explicitly show the coefficients of $x$ and $y$:
$x^{\prime }=0\cdot x+1\cdot y$
$y^{\prime }=1\cdot x+0\cdot y$

5. Representing this system of linear equations in matrix 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. Therefore, the transformation matrix for mirroring a point about the line $y=x$ is:
$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$
This matches the matrix shown in image_2.png.