The transformation $k$ is the reflection in the line $y = -x + 5$. By using the translation $h$ that maps the point $(0, 5)$ to the origin, and its inverse $h^{-1}$, determine $k$ in the form $Ax+b$.
Attempt: The idea is to translate $(0,5)$ back to origin, reflect it on $y = -x$ and then move everything back.
$$ x' = x + h = x + (0,-5)^T$$
At origin, reflect about $y=-x$ $$ x'' = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} \cdot x + (5,0)^{T} $$ Now, finally to undo the first step, add the translation $h^{-1} = (0,5)^T$. $$ \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} \cdot x + (5,0)^{T} + (0,5)^T = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} \cdot x + (5,5)^{T} $$ which is the required form of the transformation. I checked with an arbitrary point to confirm the result, but since I am beginner, I would like to know if my approach and solution are correct. Thank you.