How to find a 3x3 matrix with determinant =0 from which I can delete random column and random row to make it nonzero? 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.

$$\begin{pmatrix}1&1&-2\\1&-2&1\\-2&1&1\end{pmatrix}$$

This matrix corresponds to a linear map that maps the standard basis vectors to the column vectors of the matrix. Removing a row corresponds to restricting the linear map to the $xy$-, $xz$- or $yz$-plane. Removing a column corresponds to composing the linear map with the projection to the $xy$-, $xz$- or $yz$-plane. So it is necessary and sufficient that no two column vectors project to the same vector under any of the three projections. Equivalently, each pair of column vectors must differ in at least two coordinates.
For the matrix to have zero determinant the column vector must lie in a common plane. By the above each of the three projections, restricted to this plane, must be injective, so the plane must be given by an equation $ax+by+cz=0$ with $abc\neq0$.
Then it is geometrically clear that any such plane and any three pairwise nonparallel vectors in it will suffice as column vectors. I chose the plane $x+y+z=0$ and three obvious vectors on it.
Geometrically such matrices correspond bijectively, up to scalar multiples, to triples of distinct collinear points in the projective plane whose common line does not pass through $(1:0:0)$, $(0:1:0)$ and $(0:0:1)$.
