Please check my proof

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Prove that the reflection matrix:

[math]
S = \begin{bmatrix}
2x^{2} - 1 & 2xy\\
2xy & 2y^{2} - 1
\end{bmatrix} [/math] is of the form [math]\begin{bmatrix}
a & b \\
b & -a
\end{bmatrix} [/math] where [math]a^{2} + b^{2} = 1[/math]

Proof:

[eqn]a = -(-a) [/eqn][eqn]2x^2-1 = 1-2y^2 [/eqn][eqn]2x^2 - 2 = -2y^2 [/eqn][eqn]2(x^2 -1) = -2y^2 [/eqn][eqn]x^2 - 1 = -y^2 [/eqn][eqn]x^2 + y^2 = 1 \;\;\;\;\;\;\;\;\;\;\;\; [1][/eqn]

[eqn](2x^2 - 1)^2 + (2xy)^2 = 1 [/eqn][eqn]4x^4 - 4x^2 + 1 + 4x^2y^2 = 1 [/eqn][eqn]4x^2(x^2 - 1 + y^2) = 0 \;\;\;\;\;\;\;\;\;\;\;\; [2] [/eqn]

now plugin [1] into [2]:
[eqn]4x^2(1 - 1) = 0 [/eqn]