Answer
$T=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\end{array}\right]$
Work Step by Step
Given: points are rotated about the origin through $-\frac{\pi}{4}$ radians.
General rotation matrix for rotation about the origin through $\theta$ radians:
\[
T=\left[\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right]
\]
Replace $\theta$ by $-\frac{\pi}{4}$ and evaluate:
\[
T=\left[\begin{array}{cc}
\cos -\frac{\pi}{4} & -\sin -\frac{\pi}{4} \\
\sin -\frac{\pi}{4} & \cos -\frac{\pi}{4}
\end{array}\right]=\left[\begin{array}{cc}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
-\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}
\end{array}\right]
\]
