Answer
See solution.
Rotating by $90^o$ around origin can be done by multiplying vector $\begin{bmatrix}
0&-1\\
1&0
\end{bmatrix}$
Work Step by Step
Reflecting points through the $x_1$ then the $x_2$ axis is the same as rotating it around the origin by $180^o$.
Lets demonstrate on vector $b=\begin{bmatrix}
b_1\\
b_2
\end{bmatrix}$
a) Reflecting through $x_1$
$\begin{bmatrix}
1&0\\
0&-1
\end{bmatrix}\begin{bmatrix}
b_1\\
b_2
\end{bmatrix}=\begin{bmatrix}
b_1\\
-b_2
\end{bmatrix}$
Reflecting through $x_2$
$\begin{bmatrix}
-1&0\\
0&1
\end{bmatrix}\begin{bmatrix}
b_1\\
-b_2
\end{bmatrix}=\begin{bmatrix}
-b_1\\
-b_2
\end{bmatrix}$
Resulting vector is$\begin{bmatrix}
-b_1\\
-b_2
\end{bmatrix}$
b) Rotating twice around origin by $90^o$
$\begin{bmatrix}
0&-1\\
1&0
\end{bmatrix}\begin{bmatrix}
b_1\\
b_2
\end{bmatrix}=\begin{bmatrix}
-b_2\\
b_1
\end{bmatrix}$
$\begin{bmatrix}
0&-1\\
1&0
\end{bmatrix}\begin{bmatrix}
-b_2\\
b_1
\end{bmatrix}=\begin{bmatrix}
-b_1\\
-b_2
\end{bmatrix}$
Resulting vector is$\begin{bmatrix}
-b_1\\
-b_2
\end{bmatrix}$, which is the same as part a)
