The transformation matrix for mirroring a point in x-y plane about the line y = x is given by A. \[\left[ {\begin{array}{*{20}{c}} { – 1}&0 \\ 0&1 \end{array}} \right]\] B. \[\left[ {\begin{array}{*{20}{c}} 0&1 \\ 1&0 \end{array}} \right]\] C. \[\left[ {\begin{array}{*{20}{c}} 0&{ – 1} \\ { – 1}&0 \end{array}} \right]\] D. \[\left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&{ – 1} \end{array}} \right]\]

\]” option2=”\[\left[ {\begin{array}{*{20}{c}} 0&1 \\ 1&0 \end{array}} \right]\]” option3=”\[\left[ {\begin{array}{*{20}{c}} 0&{ – 1} \\ { – 1}&0 \end{array}} \right]\]” option4=”\[\left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&{ – 1} \end{array}} \right]\]” correct=”option1″]

The correct answer is $\boxed{\left[ {\begin{array}{*{20}{c}} { – 1}&0 \ 0&1 \end{array}} \right]}$.

A transformation matrix is a matrix that describes how a geometric object is transformed under a certain transformation. In this case, the transformation is mirroring a point in the $x$-$y$ plane about the line $y=x$.

To find the transformation matrix, we can use the following steps:

1. Find the vector that represents the line $y=x$. This vector is $(1,1)$.
2. Find the normal vector to the line $y=x$. This vector is $(-1,0)$.
3. Take the dot product of the two vectors. This gives us the determinant of the transformation matrix, which is $-1$.
4. The transformation matrix is then given by $\left[ {\begin{array}{*{20}{c}} { – 1}&0 \ 0&1 \end{array}} \right]$.

The other options are incorrect because they do not represent the correct transformation. Option A is the identity matrix, which does not change the position of any points. Option B is the matrix that represents a rotation of 90 degrees counterclockwise around the origin. Option C is the matrix that represents a reflection across the $y$-axis. Option D is the matrix that represents a reflection across the $x$-axis.