§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．