Let $A \in \mathbb{R}^{n \times n}$ be a full rank matrix. Let $B_k \in \mathbb{R}^{k \times k}$ be the leading principal submatrix of $A$ formed by taking the first $k$ rows and the first $k$ columns of $A$.
What can we say about the rank of $B_k$, for $k = 1, \cdots, n$? Is it true that $B_k$ is of full rank for all $k = 1, \cdots, n$? My intuition tells me this should be true, as, otherwise, $A$ would not be of full rank.