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  3. let a be an n × n real symmetric matrix with i

Let A be an n × n real symmetric matrix with its row and column sums...

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Advance Math

Let A be an n × n real symmetric matrix with its row and columnsums both equal to 0. Let λ1, . . . , λn bethe eigenvalues of A, with λn = 0, and withcorresponding eigenvectors v1,...,vn (theseexist because A is real symmetric). Note that vn = (1, .. . , 1). Let A[i] be the result of deleting the ith row andcolumn.

Prove that detA[i] = (λ1···λn-1)/n. Thus,the number of spanning trees of G is the product of the nonzeroeigenvalues of L, divided by n.

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