TITLE: Show that B is a nonsingular matrix (not that obvious).
QUESTION [21 upvotes]: How would you proceed if you were asked in an interview to show that B is a nonsingular matrix (in an elegant way)?
$$B= \begin{pmatrix} 1& 1.25& −0.50& 0.15\\ 0.15& 2& 1.25& −1.50\\ −0.45& 0.25& 3& 1.25\\ 0.25& −0.15& 0.25& 4\\ \end{pmatrix}$$
In my opinion, taking the time to compute the determinant of this $4\times4$ matrix during the interview would not be appreciated by the interviewer.

REPLY [2 votes]: Something that will always work with low complexity and not dependent on which matrix:

Take a random vector
Loop over columns / rows: 


Remove projection with current row/col


If after three (or $n-1$) removed projections is not parallel to last vector ( up to precision ) then there is a non-zero sized null-space.
I'm very sure that the chance of accidentally hitting a subspace in the value space is abysmally small.