302 - Proof (a) If B is obtained by interchanging row r and...

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1 Proof (a) If B is obtained by interchanging row r and row s = r +1 of A , a rj = b sj and A rj = B sj j . ( 1) r +1 a r 1 det A r 1 + ( 1) r +2 a r 2 det A r 2 + " + ( 1) r + n a rn det A rn = ( 1) r +1 b s 1 det B s 1 + ( 1) r +2 b s 2 det B s 2 + " + ( 1) r + n b sn det B sn = ( 1) s +1 b s 1 det B s 1 ( 1) s +2 b s 2 det B s 2 " ( 1) s + n b sn det B sn det A = det B . If B is obtained by interchanging row r and row s > r +1 of A , B may be obtained from A by making adjacent row interchanges: + + n s r s r s r a a a a a a a a # # # 1 1 1 1 1 + + n s s s r r r a a a a a a a a # # # 1 1 1 1 1
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302 - Proof (a) If B is obtained by interchanging row r and...

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