Answer
$A=\begin{bmatrix}
-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}
\end{bmatrix}$
Work Step by Step
$A=\begin{bmatrix}
T(e_1) & ... & T(e_n)\\
\end{bmatrix}$
$e_1=\begin{bmatrix}
1\\
0
\end{bmatrix}$ and $e_2=\begin{bmatrix}
0\\
1
\end{bmatrix}$
$T(e_1)=\begin{bmatrix}
-\frac{1}{\sqrt{2}}\\
\frac{1}{\sqrt{2}}
\end{bmatrix}$ and
$T(e_2)=\begin{bmatrix}
1&0\\
0&-1
\end{bmatrix}\begin{bmatrix}
-\sin(-\frac{3\pi}{4})\\
\cos(-\frac{3\pi}{4})
\end{bmatrix}=\begin{bmatrix}
\frac{1}{\sqrt{2}}\\
\frac{1}{\sqrt{2}}
\end{bmatrix}$
$A=\begin{bmatrix}
-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\
\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}
\end{bmatrix}$
