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22
votes
Accepted
Can such a matrix exist?
No such matrix exists.
The Desnanot-Jacobi identity says that for $i \neq j$,
$$
\det M_{ij}^{ij} \det M = \det M_i^i \det M_j^j -\det M_i^j \det M_j^i
$$
so using this, we get
$$
\det M_{12}^{12 …