If a square matrix $A$ is nonsingular, then every submatrix of $A$ is also nonsingular.
I am trying to come up with a counter example. But most involve really difficult examples, so I am starting to think this is actually true and probably something to do with a non zero determinant through each sub matrix.
Also, does taking a zero entry in say an identity count as a non singular matrix? (1 x 1 matrix)