14-240/Tutorial-December 2

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Boris

Theorem

Let be a matrix and be the matrix with two rows interchanged. Then . Boris decided to prove the following lemma first:

Lemma 1

Let be a matrix and be the matrix with two adjacent rows interchanged. Then .


All we need to show is that . Assume that is the matrix with rows of interchanged. Since the determinant of a matrix with two identical rows is , then:




.


Since the determinant is linear in each row, then we continue where we left off:




.


Then and . The proof of the lemma is complete.


For the proof of the theorem, assume that is the matrix with rows of interchanged and . By Lemma 1, we have the following:






.


Then the proof of the theorem is complete.

Nikita