Exercise 3.2.17

Let $B=\left(\begin{array}{c}\hfill b_1\hfill \\ \hfill b_2\hfill \\ \hfill b_3\hfill \end{array}\right)$ and $C=\left(\begin{array}{ccc}\hfill c_1\hfill & \hfill c_2\hfill & \hfill c_3\hfill \end{array}\right)$. Thus we know that

has only at most one independent rows. So the rank of $\mathit{BC}$ is at most one.

Conversely, if the rank of $A$ is zero, we know that $A=O$ and we can pick $B$ and $C$ such that they are all zero matrices. So assume that the rank of $A$ is $1$ and we have that the $i$-th row of $A$ forms a maximal independent set itself. This means that we can obtain the other row of $A$ by multiplying some scalar (including $0$), say $b_j$ for the $j$-the row. Then we can pick $B=\left(\begin{array}{c}\hfill b_1\hfill \\ \hfill b_2\hfill \\ \hfill b_3\hfill \end{array}\right)$ and $C=\left(\begin{array}{c}\hfill A_{i1}\hfill \\ \hfill A_{i2}\hfill \\ \hfill A_{i3}\hfill \end{array}\right)$. Thus we get the desired matrices.