TITLE: How to find a 3x3 matrix with determinant =0 from which I can delete random column and random row to make it nonzero?
QUESTION [7 upvotes]: I need to find a $3 \times 3$ matrix and the determinant of this matrix has to be $0$.
I also need to be able to delete randomly chosen column and row to make the determinant nonzero?  Is it even possible? Thank you.

REPLY [2 votes]: Pick an arbitrary nonsingular $2\times2$ matrix $P$ and also four nonzero numbers $u_1,u_2,v_1,v_2$. Let
$$
A=\pmatrix{1&0\\ 0&1\\ u_1&u_2}P\pmatrix{1&0&v_1\\ 0&1&v_2}.\tag{1}
$$
Then $A$ is singular because it has rank 2, but its every $2\times2$ submatrix is nonsingular because it is the product of three nonsingular $2\times2$ matrices. In fact, every singular $3\times3$ matrix whose all $2\times2$ submatrices are nonsingular can be written in the form of $(1)$.