Under a certain coordinate transformation from $(x, y)$ to $(u, v)$ the circle $x^{2}+y^{2}=1$ shown below on the left side was transformed into the ellipse shown on the right side.
If the transformation is of the form
\[
\left[\begin{array}{l}
u \\
v
\end{array}\right]=\mathbf{A}\left[\begin{array}{l}
x \\
y
\end{array}\right],
\]
which of the following could the matrix A be:
\[
\begin{aligned}
A_{1} & =\left[\begin{array}{ll}
a & 0 \\
0 & b
\end{array}\right]\left[\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right] \\
A_{2} & =\left[\begin{array}{ll}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right]\left[\begin{array}{ll}
a & 0 \\
0 & b
\end{array}\right] \\
A_{3} & =\left[\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right]\left[\begin{array}{ll}
a & 0 \\
0 & b
\end{array}\right]\left[\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right]
\end{aligned}
\]
- $A_{1}$ only
- $A_{2}$ only
- $A_{1}$ or $A_{2}$
- $A_{1}$ or $A_{3}$
- $A_{2}$ or $A_{3}$