Connection between Two lemmas

Fawad Hussain

Friday 23rd April, 2010 16:00-17:00 Mathematics Building, room 515


Connection between two Lemmas: In 1851, Sylvester proved the following result Lemma 1 : For any $ p \times p $ matrices M and N, and $1\leq k\leq p$, $ det(M).det(N)= \sum det(M^{'})det(N^{'})$ where the sum is over all pairs $(M^{'}, N^{'})$ of matrices from M and N by interchanging a fixed set of K columns of N with any k columns of M, preserving the ordering of the columns. One of the other Mathematician proved the following result Lemma 2 : Let R be a ring and M $\in M_n(R)$.Let $N = M_{1,n;1,n}\in M_{n-2}(R)$, then we have the following. $ det(M). det(N) = det M_{n,n}.detM_{1,1}- det M_{n,1}.det M_{1,n}$ I will prove the second lemma from first lemma.

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