§14行列式的性质 性质1设D ,则DT=D 证令b=an(=12, ∑(-1)bnbn…bm,r=(PP2…Pn) ∑(-1)a D 根据Th (P1P2…Pn) 性质2设i<j,D= D ,则D1=-D 1≠i,j:h=ak(k=1,2,…,m) D b ∑(-1)(-1X…bm…b t(…P…P2…) =(-1)2(-1)(…am…am…) qi =Pj,j=Pi 1,J:q=P1 (-1)∑(-1)(…an…amn…) t(…q1…q…) 推论1D对调两列得D2→D2=-D7 §1.4 行列式的性质 性质 1 设 n nn n a a a a D 1 11 1 = , n nn n a a a a D 1 11 1 Τ = , 则 D = D Τ . 证 令 b a (i, j 1,2, ,n) ij = ji = , 则 n nn n b b b b D 1 11 1 Τ = n n p p np p p p b b b 1 2 1 2 1 2 ( ) = (−1) ( ) p1 p2 pn = ap ap ap n D p p p n n = − = 1 2 ( ) 1 2 1 2 ( 1) (根据 Th2) 性质 2 设 j jn i in a a a a i j D 1 1 , = , i in j jn a a a a D 1 1 1 = , 则 D1 = −D . 证 b a , b a (k 1,2, ,n) ik = jk jk = ik = l i, j : b a (k 1,2, ,n) lk = lk = ( 1) ( ) 1 1 1 i j i p j p j j n i i n b b b b b b D = = − ( ) pi p j ( 1) ( 1)( ) j i jp ip t = − − b b ( ) p j pi t ( 1) ( 1) ( ) j i ip jp t = − − a a l l i j j i l i j q p q p q p = = = , : , ( 1) ( 1) ( ) i j iq jq t = − − a a = −D ( ) qi qj t 推论 1 D 对调两列得 D2 D2 = −D.