Exercise 4.8.2

We can have $A_4=\left[\begin{array}{ccccc}\hfill A_{11}\hfill & \hfill A_{12}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill A_{21}\hfill & \hfill \hfill & \hfill A_{23}\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill A_{32}\hfill & \hfill \hfill & \hfill \hfill & \hfill A_{35}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill A_{44}\hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill A_{53}\hfill & \hfill \hfill & \hfill A_{55}\hfill \end{array}\right]$

To have a completion to a rank-1 matrix, we must have the entries of $A_4$ to have determinant of 0 for any square submatrix larger than 1.