Should a reflection matrix of a vector have the same form as a rotation matrix? According to the book:

I know that it is not possible to write a reflection as a rotation, but from the text it seems that the matrix of the form 
$$ A=\left[
  \begin{array}{ c c }
     a & -b \\
     b & a
  \end{array} \right]
$$
is the most general form of $O(2)$ since the way it results in $A$ doesn't show any specification of rotation; on the other hand, I can't find any pair of $a,b$ such that
$$ A=\left[
  \begin{array}{ c c }
     1 & 0 \\
     0 & -1
  \end{array} \right].
$$ 
Why is that?
Thank you. 
 A: All reflections in the plane have matrices of the form
$$
\left(
\begin{array}{cc}
\cos \alpha & \sin \alpha \\
\sin \alpha & - \cos \alpha
\end{array}
\right)
$$
or, for any $a^2 + b^2 = 1,$
$$
\left(
\begin{array}{cc}
a & b \\
b & - a
\end{array}
\right)
$$
A: HINT: $a^2+b^2=1$ and $T(e_1) =(a,b)^t$, give you two solutions not one: $T_1(e_2) =(-b,a)^t$ and $T_2(e_2) =(b,-a)^t$.
A: on the other hand, I can't find any pair of a,b such that...Why is that?
You cannot find such a pair because your assumption
but from the text it seems that the matrix of the form ..is the most general form of O(2) 
is not true: As you noted every rotation
$$
R=\left[
\begin{array}{cc}
a & -b \\
b & a
\end{array}
\right]
$$
has determinant $\det(R)=a^2+b^2=+1$ but the typical reflection
$$
A=\left[
  \begin{array}{ c c }
     1 & 0 \\
     0 & -1
  \end{array} \right].
$$
has determinant $-1$. 
Note that the most general form of an Element $A$ of $O(2)$ maps $e_1=(1,0)^t$ to $(a,b)^t$ where $a^2+b^2=1$ (length is preserved) and because $Ae_2$ must be orthogonal you have
$$
A=\left[
\begin{array}{cc}
a & -b \\
b & a
\end{array}
\right]
\quad \text{or }\quad
A=\left[
\begin{array}{cc}
a & b \\
b & -a
\end{array}
\right]
$$
