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Find a field F and matrices $A,B, C, D \in F^{n,n}$ such that

$$ \text{det}\bigg(\left[ \begin{array}{ccc} A & B \\ C & D \end{array} \right]\bigg) \neq \text{det}(A) \ \text{det}(D) - \text{det}(B) \ \text{det}(C)$$

I thought to consider if A is invertible and $AC = CA$, or other cases, but these formulars are all driven from the one above.

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When in doubt, look for a matrix that has mostly zeros. The following example applies to any field, so taking $\Bbb F = \Bbb R$ is certainly okay: $$ M = \left[ \begin{array}{cc|cc} 0&0&0&1\\ 1&0&0&0\\ \hline 0&1&0&0\\ 0&0&1&0 \end{array} \right] $$

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