方阵的逆矩阵满足下列运算规律: (i)若A可逆,则A-1可逆,且(A-1)-1=A 由逆矩阵的唯一性知AA-=E→A-1的逆矩阵必为A.A与A-1互为 31矩阵 逆矩阵。 §2矩阵的运算 53逆矩阵 (i)若A可逆,数入≠0,则λA可逆,且(A4)-1=A1 §4矩阵分块法 本章总结 证因为A≠0,A≠0,所以AA=入A≠0=→入A可逆 (λA)-1(4)=E=(A)-1A=E=(A)1A=1E 主讲:张少 (AA)-AA-=JEA=(A)-=JA-1 标题页 (ⅲi)若A和B为同阶方阵且均可逆,则AB也可逆,且(AB)-1=B-1A-1 44 证(AB)(B-1A-1)=A(BB1)A-1=AEA-1=A-1=E (AB)-1=B-1A1 第11页共36页 注:依次类推,A1,A2,An均可逆,则有(A142…An)-1 A-A AoA 全屏显示 (iy)若A可逆,则A亦可逆,且(A1)-1=(A-1) A(A-1)=(A-14)T=E=E=→(A-1)T=(A-1)天津师范大学 §1 Ý ✡ §2 Ý ✡ ✛ ✩ ➂ §3 ❴ Ý ✡ §4 Ý ✡ ➞ ➡ ④ ✢Ù♦✭ ❒ù: Ü✟r ■ ❑ ➄ JJ II J I ✶ 11 ➄ ✁ 36 ➄ ❼ ↔ ✜ ➯ ✇ ➠ ✬ ✹ ò Ñ ➄✡✛❴Ý✡÷✈❡✎✩➂✺➷➭ (i) ❡A➀❴➜❑A−1➀❴➜❹(A−1 ) −1 = A. ❞❴Ý✡✛➁➌✺⑧AA−1 = E =⇒ A−1 ✛❴Ý✡✼➃A. A❺A−1♣➃ ❴Ý✡✧ (ii) ❡A➀❴➜êλ 6= 0, ❑λA➀❴➜❹(λA) −1 = 1 λA−1 . ② Ï➃|A| 6= 0, λ 6= 0, ↕➧|λA| = λ n |A| 6= 0 =⇒ λA ➀❴✧ (λA) −1 (λA) = E =⇒ λ(λA) −1A = E =⇒ (λA) −1A = 1 λE =⇒ (λA) −1AA−1 = 1 λEA−1 =⇒ (λA) −1 = 1 λA−1 . (iii) ❡AÚB➃Ó✣➄✡❹þ➀❴➜❑AB➃➀❴➜❹(AB) −1 = B−1A−1 . ② (AB)(B−1A−1 ) = A(BB−1 )A−1 = AEA−1 = AA−1 = E =⇒ (AB) −1 = B−1A−1 . ✺ ➭ ➑ ❣ ❛ í ➜A1, A2, . . . An þ ➀ ❴ ➜ ❑ ❦(A1A2 · · · An) −1 = A−1 n A −1 n−1 · · · A −1 2 A −1 1 . (iv) ❡A➀❴➜❑AT➼➀❴➜❹(AT) −1 = (A−1 ) T. ② AT(A−1 ) T = (A−1A) T = E T = E =⇒ (A−1 ) T = (A−1 ) T.