What information do you gain by knowing that all $k \times k$ minors of a $n \times m$ matrix vanish? So firstly, these kinds of properties were not directly addressed in the linear algebra course I have taken, so there might be some "language barrier" holding me back. In any case, I presume that a $k \times k$ minor refers to the value of the $k \times k$ determinant. Given the knowledge that all $k \times k$ minors of a matrix $A$ are zero, what else do we know about the matrix? I honestly do not know, as besides performing manual calculations, the only "theory" and practise behind determinants that I have encountered has been the recursive definition with subdeterminant rule(s).
 A: The $k \times k$ minors vanish if and only if the rank of $A$ is less than $k$, as Greg says in the comments. Here's a relatively low-tech way to see this.
A matrix $A$ has rank less than $k$ iff the dimension of the column space $\text{col}(A)$ is less than $k$. The dimension of the column space is the maximum size of a linearly independent set of columns, so to check this condition it suffices to check whether every $k$-tuple of columns is linearly dependent. In other words, to check that a matrix $A$ has rank less than $k$ it suffices to check whether all the matrices obtained by selecting $k$ columns of $A$ (and ignoring the rest) have rank less than $k$.
But the dimension of the column space is the dimension of the row space, so exactly the same argument applies to the rows of $A$. If we apply the argument to the columns first, producing a bunch of matrices with $k$ columns, and then apply the argument to the rows of these matrices, we conclude that a matrix $A$ has rank less than $k$ iff all the $k \times k$ submatrices obtained by selecting $k$ columns and $k$ rows of $A$ (and ignoring the rest) have rank less than $k$. But since these are $k \times k$ matrices they have rank less than $k$ iff their determinant vanishes.
A higher-tech way to see this result is to show that the $k \times k$ minors appear as the coefficients of the exterior power of $A$, and to show that $A$ has rank less than $k$ iff its $k^{th}$ exterior power vanishes. Developing the theory of exterior powers tells you much more than this about the minors, for example it tells you they transform in a particular way under change of coordinates and that there's a formula for the minors of $AB$ in terms of the minors of $A$ and $B$, neither of which is otherwise clear.
