Discrete Mathematics : The Abelian Sandpile Model


Exercise 7. Let A be an n × n matrix such that the sum of every row is 0 and the sum of every column is 0. Let Aij be the (n − 1) × (n − 1) matrix obtained by removing row i and column j from A. Prove: det(Aij) = (−1) det(A11). (Note that this result applies in particular to the Laplacian L: the determinant of the reduced Laplacian, obtained by removing the… (More)


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