Answer
$T=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]$
Work Step by Step
We are looking for a $2 \times 2$ matrix that reflects points through the horizontal axis and then reflects points through the line $x_{2}=x_{1}$
First the points need to be reflected through the horizontal $x_{1}$ -axis. This then means that the $x_{1}$ -coordinate remains unaffected and the $x_{2}$ -coordinate changes sign:
\[
R_{1}=\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]
\]
Note: if you multiply this matrix by the vector $\left(x_{1}, x_{2}\right)^{T}$ to the right, then you get the vector $\left(x_{1},-x_{2}\right)^{T}$
Next the points need to be reflected through the line $x_{2}=x_{1}$. The $x_{1}$ -coordinate and $x_{2}$ -coordinate then interchange.
\[
R_{2}=\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right]
\]
Obseeve: if you multiply this matrix by the vector $\left(x_{1}, x_{2}\right)^{T}$ to the right, then you get the vector $\left(x_{2}, x_{1}\right)^{T}$
The combination of the reflections is then the product of the reflection matrices, with the first reflection matrix to the right.
\[
T=R_{2} \times R_{1}=\left[\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}\right] \times\left[\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right]=\left[\begin{array}{cc}
0 & -1 \\
1 & 0
\end{array}\right]
\]
Observe: if you multiply this matrix by the vector $\left(x_{1}, x_{2}\right)^{T}$ to the right, then you obtain the vector $\left(-x_{2}, x_{1}\right)^{T}$
