[Math] Let $A$ be the reflection of the plane $\mathbb R^2$ in the line $y=-x$. Find the matrix of $A$ in the standard basis $S=\{e_1,e_2\}$

Question

Let $A$ be the reflection of the plane $\mathbb R^2$ in the line $y=-x$. Find the matrix of $A$ in the standard basis $S=\{e_1,e_2\}$

I feel like my answer is too simple because the question is out of $10$ marks. Am I missing something?

$A(x,y) = (-y,-x)$

$A(e_1) = A(1,0) = (0,-1)$

$A(e_2) = A(0,1) = (-1,0)$

$\begin{bmatrix} A \end{bmatrix}_S = \begin{bmatrix} 0 & {-1} \\ {-1} & 0 \end{bmatrix}$

Answer 1

Your answer is completely correct, this is indeed a very easy problem.