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Consider a $2 \times 4$ matrix $A = \begin{bmatrix} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ \end{bmatrix}$. Its all minors of order 2, such as $A_{13} = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \\ \end{vmatrix}$ representing the minor formed by selecting the first and third columns, satisfy the following identity, which is easy to verify: $$ A_{13}A_{24}=A_{12}A_{34}+A_{14}A_{23} $$ And the identity holds in a more general situation, such as an $n \times m$ matrix where $m \geq n+2$. By selecting any fixed $n-2$ columns, and then selecting 4 distinct columns from the remaining ones using similar indices as described above, the identity still holds.

From Wikipedia, the form of the identity is the same as the Plücker relation in the case of $\mathbf{Gr}(2,4)$. From my professor, these two equations are indeed related in some sense, and I wonder where this connection comes from.

Furthermore, I'm seeking how to prove this identity without knowledge related to Grassmannian manifolds (as I know nearly nothing about it).

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  • $\begingroup$ This paper has a straightforward proof and a rough outline of the wider theory. $\endgroup$
    – Mo Pol Bol
    Feb 28 at 12:39

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