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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?

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up vote 3 down vote accepted

$A=\left[\begin{array} \\ 1&3\\ 1&2\\ \end{array}\right]\quad B=\left[\begin{array}\\ 2&4\\ 2&1\\ \end{array}\right]\quad C=\left[\begin{array} \\ 1&0\\ 1&5\\ \end{array}\right]\quad D=\left[\begin{array} \\ 2&0\\ 2&6\\ \end{array}\right]\\ |AD-BC|=20 \\ |AD-CB|=102\\ |DA-BC|=18\\ |DA-CB|=8$

but the combined matrix has one row twice another, so has a determinant of 0.

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