Complete the matrix through a rank-one approximation.

Question

Complete the matrix through a rank-one approximation:

\begin{pmatrix}1&2\\3&?\end{pmatrix}

I am new to these types of problems, therefore a detailed explanation would really be appreciated.

Answer 1

You are asking to find a matrix $X$ such that $X_{11} = 1$, $X_{12} = 2$, $X_{21} = 3$, and $\operatorname{rank} X = 1$.

Hint: What do the vectors $(1,2)$ and $(3,6)$ have in common? (drawing these vectors might help)