= Solution
The two columns are <orthogonal>, with lengths one and two; equivalently $A^TA=\operatorname{diag}(1,4)$. Ordering the <singular values> decreasingly requires swapping the right coordinate directions. One full <singular value decomposition> is
$$
\boxed{U=\begin{pmatrix}1&0&0\\0&0&1\\0&-1&0\end{pmatrix},\quad
\Sigma=\begin{pmatrix}2&0\\0&1\\0&0\end{pmatrix},\quad
V=\begin{pmatrix}0&1\\1&0\end{pmatrix}.}
$$
Both $U$ and $V$ are <orthogonal matrices>, and direct <matrix multiplication> gives $U\Sigma V^T=A$. Hence
$$
\boxed{\|A\|_2=2,\qquad\|A\|_F=\sqrt5.}
$$
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