Answer
The angle of rotation is $\frac{\pi}{2}$ or 90 deg.
Work Step by Step
\[
A=\left(\begin{array}{cc}
0 & -1 \\
1 & 0
\end{array}\right)
\]
From exercise 8 we know what the transformation
does, and we have the standard matrix $A$, for $T$.
2
\[
\begin{array}{l}
T\left(\mathbf{e}_{1}\right)=\mathbf{e}_{1} \\
T\left(\mathbf{e}_{2}\right)=-\mathbf{e}_{1}
\end{array}
\]
We can show that the standard unit vector $\mathbf{e}_{1},$ is mapped to e $_{2},$ and that the standard unit vector $\mathbf{e}_{2},$ is mapped to $-\mathbf{e}_{1} .$ Both of these mappings identical to a rotation of $\frac{\pi}{2}$ or 90 deg about the origin of each component which is the angle of the rotation.
