= Solution
For the first <matrix> choose its single independent column:
$$
\boxed{B_1=\begin{pmatrix}1\\1\\1\end{pmatrix},\qquad
C_1=\begin{pmatrix}1&1&1\end{pmatrix}.}
$$
Their product has every column equal to $B_1$, and both factors have rank one.
For the second <matrix>, the first and third columns are independent and the second equals the first, giving
$$
\boxed{B_2=\begin{pmatrix}1&0\\2&0\\0&1\end{pmatrix},\qquad
C_2=\begin{pmatrix}1&1&0\\0&0&1\end{pmatrix}.}
$$
Multiplication reconstructs the stated <matrix>. The first factor has independent columns and the second has independent rows, so both have <matrix rank> two, as required for a <rank factorization>.
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