001 0 k k000 4.矩阵的转置 a A aln a2 算律:(1)(A)=A(2)(Amn+Bnn)=A+B (3)(kA)=kA(4)(Am,B,n)2=BA 验证(4)A=(an)m,B=(b) AB=C=(Cumxm, B A=D=(du)mxmt [=cn=pn…a,l:=a+…+a. [右=d= bn…b b, a b 故d=cn(=12,…,n,j=12,…m),即(AB)2=BA 对称矩阵:指A满足A=A,即an=an(i,j=12,…,m) 反对称矩阵:指Am满足A=-A,即a=-an(,j=12,…,m 5.方阵的行列式:指A=(an)mn的元素按照原来的相对位置构成的 行列式,记作det4,或者1 算律:(1)detf=detd (2)det(la)=/" detA (3)det(AB)=(det a)(detb)(4)detA"=(det a)6           =           +           = 1 2 0 1 0 0 0 0 0 0 0 0 0 1 1 2 1 k k k k 4. 矩阵的转置：             = m m mn n n a a a a a a a a a A       1 2 21 22 2 11 12 1 ,             = n n mn m m a a a a a a a a a A       1 2 12 22 2 11 21 1 T 算律：(1) A = A T T ( ) (2) T T T (Amn + Bmn ) = A + B (3) T T (k A) = k A (4) T T T (AmsBsn ) = B A 验证(4) A = aij ms ( ) , B = bij sn ( ) ij m n AB C c = =  ( )  , B A =D = dij nm ( ) T T  左ij =   j i j s si si i ji j j s a b a b b b c a a = + +           =   1 1  1 1 右ij =   i j si j s j i j s j i j i si b a b a c a a d b b = + + =           =   1 1  1 1 故 d c (i 1,2, ,n; j 1,2, ,m) ij = ji =  =  ，即 T T T (AB) = B A ． 对称矩阵：指 Ann 满足 A = A T ，即 a a (i, j 1,2, ,n) ij = ji =  反对称矩阵：指 Ann 满足 A = −A T ，即 a a (i, j 1,2, ,n) ij = − ji =  5. 方阵的行列式：指 A = aij nn ( ) 的元素按照原来的相对位置构成的 行列式, 记作 detA, 或者 A ． 算律：(1) detA detA T = (2) l A l A n det( ) = det (3) det(AB) = (detA)(detB) (4) k k detA = (detA)