TITLE: Block Determinants
QUESTION [6 upvotes]: 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?

REPLY [4 votes]: $A=\left[\begin{array}  \\
1&3\\
1&2\\
\end{array}\right]\quad
B=\left[\begin{array}\\
2&4\\
2&1\\
\end{array}\right]\quad
C=\left[\begin{array}  \\
1&0\\
1&5\\
\end{array}\right]\quad
D=\left[\begin{array} \\
2&0\\
2&6\\
\end{array}\right]\\
|AD-BC|=20 \\
|AD-CB|=102\\
|DA-BC|=18\\
|DA-CB|=8$
but the combined matrix has one row twice another, so has a determinant of 0.