第十二讲冲刺篇 一重要公式与结论 (一，行列式与矩阵 1.kA川=k叫A:4-1=A-11AB=ABL,其中A为n阶方阵. 2.若A=(ag)且r(A=1,则A=a87,其中a=(a1,…,an)P,B=(b,…,b)T,且 (i)BTa =tr(A)=au+azz+...+amn: (间AE-A=X-(a1+…+anbnA=n-(a1+…+anmA 3.A1=☆A:(kA)1=是A1(AB)-1=B-1A: 4.A"A=A4°=4E:A=4-1;(m≥2：(kA)=-1A:(4)-1=☆4，A°=144-1，其 中A为n阶方阵. sA-(:)则-(）4递点(。) 6.E6》=分，G什GE()=n×k(G×kE6,》=n+乃×k(G+马× 7.Ei,》1=E6,),E(》1=E(使》，E6,k》-1=E,-k》.特别地，E6,P=E 8.⑤)A行元B=E,)4=B.A石dB=AE6,)=B: 回AXB=EA=B.Aoxk B电AE》二E.=B A+B=E)A=B, 9.()若A可逆，将(A,E)行变损(E,X),则X=A-1. (2)①)若AX=B且A可逆，将(A,B)行变换(E,X),则X=A-1B. (问)若xA=B且A可逆，则ATXT=BT,将(AT,B行变孩(E,x),则xT=(A-BT,从 而X=BA (仁)，向量与线性方程组 1.3可以由向量组a1,a2, ,a的线性表出 台存在一组数，，，k使得1+十十ka,=月 台方程101+工202十…+工，a,=有解(1,2，…，工，T=(k1,2,…,k,) 2.可以由向量组a1,a2,…,a,的唯一线性表出（或两种以上线性表出，不能线性表出） 台方程x1a1+x202+…+x,a,=8有唯一解（无穷多，无解） ÷r(B)=r(A)=s(或r(B=r(A)<s,r(A)<r(B). 其中A=(a1,a2,…,a),B=(4,). 3.向量组B=(G3,…,月)可以由A=(a1,a2,…,am)线性表示 ÷方程AX=B有解台r(A,B)=r(A: 4.向量组A=(a1,2,·,m)与B=(问，2，…，Bn)等价台r(A,B)=r(A)=r(B). 11õ˘ ¿eü ò ­á˙™Ü(ÿ (ò), 1™Ü› 1. |kA| = k n|A|; |A−1 | = |A| −1 ; |AB| = |A||B|, Ÿ•Aènê . 2. eA = (aij )Ör(A) = 1, KA = αβT , Ÿ•α = (a1, · · · , an) T , β = (b1, · · · , bn) T , Ö (i) β T α = tr(A) = a11 + a22 + · · · + ann; (ii) |λE − A| = λ n − (a1b1 + · · · + anbn)λ = λ n − (a11 + · · · + ann)λ. 3. A−1 = 1 |A|A∗ ; (kA) −1 = 1 kA−1 ; (AB) −1 = B−1A−1 ; 4. A∗A = AA∗ = |A|E; |A∗ | = |A| n−1 ; (n ≥ 2); (kA) ∗ = k n−1A∗ ; (A∗ ) −1 = 1 |A|A, A∗ = |A|A−1 , Ÿ •Aènê . 5. eA = a b c d ! , KA∗ = d −b −c a ! , Aå_, A−1 = 1 ad−bc d −b −c a ! . 6. E(i, j) ri ↔ rj (ci ↔ cj );E(i(k)) ri × k (ci × k); E(i, j(k)) ri + rj × k (ci + cj × k); 7. E(i, j) −1 = E(i, j), E(i(k))−1 = E(i(( 1 k )), E(i, j(k))−1 = E(i, j(−k)). AO/,E(i, j) 2 = E. 8. (i) A −−−−→ ri ↔ rj B E(i, j)A = B, A −−−−→ ci ↔ cj B AE(i, j) = B; (ii) A −−−→ ri × k B E(i(k))A = B, A −−−→ ci × k B AE(i(k)) = B; (iii) A −−−−−−−→ ri + rj × k B E(i, j(k))A = B, A −−−−−−−→ ci + cj × k B AE(j, i(k)) = B. 9. (1) eAå_, Ú(A, E) −−−−→ 1CÜ (E, X), KX = A−1 . (2) (i) eAX = BÖAå_, Ú(A, B) −−−−→ 1CÜ (E, X), KX = A−1B. (ii) eXA = BÖAå_, KAT XT = BT , Ú(AT , BT ) −−−−→ 1CÜ (E, XT ), KXT = (AT ) −1BT , l X = BA−1 . (), ï˛ÜÇ5êß| 1. βå±dï˛|α1, α2, · · · , αsÇ5L— ⇔ 3ò|Ík1, k2, · · · , ks¶k1α1 + k2α2 + · · · + ksαs = β ⇔êßx1α1 + x2α2 + · · · + xsαs = βk)(x1, x2, · · · , xs) T = (k1, k2, · · · , ks) T . 2. βå±dï˛|α1, α2, · · · , αsçòÇ5L—(½¸´±˛Ç5L—,ÿUÇ5L—) ⇔êßx1α1 + x2α2 + · · · + xsαs = βkçò)(ðı,Ã)) ⇔ r(B) = r(A) = s(½r(B) = r(A) < s, r(A) < r(B)). Ÿ•A = (α1, α2, · · · , αs), B = (A, β). 3. ï˛|B = (β1, β2, · · · , βn)å±dA = (α1, α2, · · · , αm)Ç5L´ ⇔êßAX = Bk)⇔ r(A, B) = r(A); 4. ï˛|A = (α1, α2, · · · , αm)ÜB = (β1, β2, · · · , βn)d⇔r(A, B) = r(A) = r(B). 1