(1)A的逆矩阵是唯一的，记为A-1.因此AB=BA=E,则B=A-1: (②矩阵A可逆=4A≠0，A-1=向. 3.矩阵可逆的性质及判断 )n阶方阵A可逆 =存在n阶矩阵B使得AB=BA=E(定义) =存在n阶矩阵B使得AB=E =A≠0. (田)若A,B可逆，数k≠0.则 )A=N (⑤)4-1=4-1(6)(4)-1=(4-1). ()若AB=AC,且A可逆，则B=C 4.关于伴随矩阵的运算规律 (1)4A=A4*=4E:(241=14n-1:(n≥2：(3)(4）=4m-24: (④)(kA=k-1A:(6)若A可逆，则(A)1=A,A=4A-1 5.求伴随矩阵，转置和求逆矩阵三种运算之间的关系 ()(4T”=(4T(②（4)-1=(4-1，其中A可逆（③）(4)-1=(4-1”，其中A可逆. 注意伴随矩阵A“,它由A的代数余子式所构成，基本关系式为AA=A“A=AE,求逆，转置，伴随三 个运算能交换次序， 101 (2)设A,B为3阶矩阵，E为3阶单位矩阵，满足A2B-A-B=E,A= 020 ,则B=() -201 101 (3)设A,B为3阶矩阵，A为A的伴随矩阵，E为3阶单位矩阵，满足ABA“=2BA"+E,A= 020 -201 则B=(方 100 (④)设A,B为3阶矩阵，A为A的伴随矩阵，满足A1BA=6A+BA,A= 020 ,则B=(方 003 (⑤)设A为3阶矩阵，A为A的伴随矩阵，4=，则（A)1-8A1=( 例7(1)）设A=(a)为3阶矩阵，且A*=AT,a1=a2=a13=a>0,则a=() 111 (②)设A= 022 ,(A是A1的伴随矩阵，则(4=() 003 例8()设A是n(n≥3)阶方阵，A“是其伴随矩阵，又k为常数，且k≠0，士1则必有(kA)~= 5 (1) A_› ¥çò,PèA−1 . œdAB = BA = E,KB = A−1 ; (2) › Aå_ |A| 6= 0,A−1 = 1 |A|A∗ . 3. › å_5ü9‰ (i) nê Aå_ 3n› B¶AB = BA = E(½¬) 3n› B¶AB = E |A| 6= 0. (ii) eA, Bå_, Ík 6= 0. K (1) A−1 = 1 |A|A∗ (2) A−1èå_,Ö(A−1 ) −1 = A (3) (kA) −1 = 1 kA−1 (4) (AB) −1 = B−1A−1 , Ì2(A1A2 · · · An) −1 = A−1 n · · · A −1 2 A −1 1 , Ÿ•A1, A2, · · · , Anå_. (5) |A−1 | = |A| −1 (6) (An) −1 = (A−1 ) n. (iii) eAB = AC, ÖAå_, KB = C. 4. 'uäë› $é5Æ (1)A∗A = AA∗ = |A|E; (2)|A∗ | = |A| n−1 ; (n ≥ 2); (3)(A∗ ) ∗ = |A| n−2A; (4)(kA) ∗ = k n−1A∗ ; (5)eAå_,K(A∗ ) −1 = 1 |A|A, A∗ = |A|A−1 . 5. ¶äë› ,=ò⁄¶_› n´$éÉm'X (1) (AT ) ∗ = (A∗ ) T (2) (A∗ ) −1 = (A−1 ) ∗ , Ÿ•Aå_ (3) (A∗ ) −1 = (A−1 ) ∗ , Ÿ•Aå_. 5ø äë› A∗ ,ßd|A|ìÍ{f™§§,ƒ'X™èAA∗ = A∗A = |A|E,¶_,=ò,äën á$éUÜgS. ~6 (1) A = 2 1 −1 2 ! , Eè2¸†› , B˜vBA = B + 2E, K|B| = ( ); (2) A, Bè3› , Eè3¸†› ,˜vA2B − A − B = E, A =   1 0 1 0 2 0 −2 0 1  , K|B| = ( ); (3) A, Bè3› ,A∗ èAäë› , Eè3¸†› ,˜vABA∗ = 2BA∗+E, A =   1 0 1 0 2 0 −2 0 1  , K|B| = ( ); (4) A, Bè3› ,A∗ èAäë› , ˜vA−1BA = 6A+BA, A =   1 0 0 0 2 0 0 0 3  , K|B| = ( ); (5) Aè3› ,A∗ èAäë› , |A| = 1 8 ,K( 1 3A) −1 − 8A∗ | = ( ); ~7 (1) A = (aij )è3› , ÖA∗ = AT , a11 = a12 = a13 = a > 0, Ka = ( ). (2) A =   1 1 1 0 2 2 0 0 3   , (A−1 ) ∗¥A−1äë› , K(A−1 ) ∗ = ( ) ~8 (1) A¥n(n ≥ 3)ê ,A∗¥Ÿäë› ,qkè~Í,Ök 6= 0, ±1 K7k(kA) ∗ = 5