This is a nice question I recently found in Golan's book.
Problem: Let $A,B,C,D$ be $n\times n$ matrices over $\mathbb{R}$ with $n\ge 2$, and let $M$ be the $2n\times 2n$ matrix \begin{bmatrix} A & B \\ C & D\\ \end{bmatrix}
If all of the "formal determinants" $AD-BC$, $AD-CB$, $DA-CB$, and $DA-BC$ are nonsingular, is $M$ necessarily nonsingular? If $M$ is nonsingular, must all of the formal determinants also be nonsingular?