同济大学线性代数答案详解.pdf

1 SK1-1 1. A = (A,b) A = 1122 2131 1114 ,b = 1 3 5 dk 1122 2131 1114 x1 x2 x3 = 1 3 5 ?5|?/ x1+ x2+ 2x3+ 2x4= 1, 2x1+ x2+ 3x3 x4= 3, x1 x2+ x3+ 4x4= 5. 2. 12 ab ! + xy 34 ! = 34 71 ! 1 + x2 + y a + 3b + 4 ! = 34 71 ! ? 1 + x = 3 2 + y = 4 a + 3 = 7 b + 4 = 1 kx = 2,y = 6,a = 4,b = 3. 3. 1 (1) A + 2B = 312 212 ! + 2 151 210 ! = 3 + 21 + 102 + 2 2 41 22 ! = 594 212 ! 3A B = 3 312 212 ! 151 210 ! = 9 13 56 1 6 + 23 + 16 ! = 885 846 ! (2) ABT= 312 212 ! 12 51 10 = 3 5 + 26 + 1 2 + 5 24 1 ! = 05 55 ! ATB = 32 11 22 151 210 ! = 3 415 23 1 25 11 2 + 410 + 22 = 1133 361 6122 2 4. (A + B)(A B) = 13 12 ! + 20 31 ! 13 12 ! 20 31 ! = 33 41 ! 13 23 ! = 3 + 69 9 4 212 + 3 ! = 318 69 ! 5. A2+ 3A 2B = 100 012 021 2 + 3 100 012 021 2 100 025 052 = 100 054 045 + 300 036 063 + 200 0410 0104 = 1 + 3 200 05 + 3 44 + 6 10 04 + 6 105 + 3 4 = 200 040 004 6. (1) 1 2 3 ? 321 ? = 321 642 963 (2) ? 231 ? 1 1 2 = 2 3 + 2 = 1 3 (3) ? xyz ? 111 121 110 x y z = ? x + y zx + 2y + zx + y ? x y z = (x2+ xy xz) + (xy + 2y2+ yz) + (xz + yz) = x2+ 2y2+ 2xy 2xz + 2yz (4) ? B = 010 001 000 K A = I + B = 110 011 001 ?k 2,Bk= 0 An= (I + B)n= I + nB + n(n 1) 2 B2+ 0 = 100 010 001 + 0n0 00n 000 + 00 n(n1) 2 000 000 = 1n n(n1) 2 01n 001 7.A?BvAB = BA, ? B = ab cd ! 4 =k 11 01 ! ab cd ! = ab cd ! 11 01 ! a + cb + d cd ! = aa + b cc + d ! d a + c = a b + d = a + b c = c d = c + d ( c = 0 a = d A? B = ab 0a ! , a,b?. 8. y (AT+ A)T= A + AT= AT+ A (AT A)T= A AT= (AT A) AT+ A?,AT A?. 9. An= 1 2 . n n = n 1 n 2 . n n SK1-2 1. A = 3100 2100 0014 0025 = A1O OA2 ! 5 B = 1010 0101 3021 1112 = B1B2 B3B4 ! C = 2400 1300 0031 0002 = C1O OC2 ! AC = A1O OA2 ! C1O OC2 ! = A1C1O OA2C2 ! ? A1C1= 31 21 ! 24 13 ! = 715 511 ! ,A2C2= 14 25 ! 31 02 ! = 39 612 ! AC = 71500 51100 0039 00612 AB BTA = A1O OA2 ! B1B2 B3B4 ! BT 1 BT 3 BT 2 BT 4 ! A1O OA2 ! = A1B1A1B2 A2B3A2B4 ! BT 1A1 BT 3A2 BT 2A1 BT 4A2 ! = A1B1 BT 1A1 A1B2 BT 3A2 A2B3 BT 2A1 A2B4 BT 4A2 ! 6 ? A1B1 BT 1A1= 31 21 ! 10 01 ! 10 01 ! 31 21 ! = 31 21 ! 31 21 ! = 00 00 ! A1B2 BT 3A2= 31 21 ! 10 01 ! 31 01 ! 14 25 ! = 31 21 ! 517 25 ! = 216 46 ! A2B3 BT 2A1= 14 25 ! 30 11 ! 10 01 ! 31 21 ! = 74 115 ! 31 21 ! = 45 96 ! A2B4 BT 4A2= 14 25 ! 21 12 ! 21 12 ! 14 25 ! = 69 912 ! 413 514 ! = 24 42 ! AB BTA = 00216 0046 4524 9642 7 2. A = a11a12a13 a21a22a23 a31a32a33 = ? A1A2A3 ? B = 1 2 3 = ? 123 ? AB = ? A1A2A3 ? 123 ? = A11+ A22+ A33 = 1a112a123a13 1a212a223a23 1a312a323a33 3. A = ? ene1e2en1 ? duAek?uA?1k?,?k 6= 1,Aek= ek1,Ae1= en. ?i k,Akei= Ak1ei1= = Aeik+1= eik, ?i k,Akei= Aki+1Ai1ei= Aki+1e1= AkiAe1= Akien= enk+i. d Ak= AkE = Ak ? e1ekek+1en ? = ? Ake1AkekAkek+1Aken ? = ? enk+1ene1enk ? = 0Enk Ek0 ! . 4. Dk= A1OO OA2O . . . . . . . . OOAs k = Ak 1 OO OAk 2 O . . . . . . . . OOAk s SK1-3 8 1. (1)?1C?z1/? 2202 0111 1210 2531 1 2r1 r3+(1)r1 r4+(2)r1 1101 0111 0111 0333 r3+(1)r2 r4+(1)r2 1101 0111 0000 0000 r1+(1)r2 1012 0111 0000 0000 (2)?1C?z1/? 0111 0231 0471 0343 r2+(2)r1 r3+(4)r1 r4+(3)r1 0111 0053 00113 0076 1 5r2 r1+(1)r2 r3+11r2 r4+7r2 010 2 5 0013 5 00018 5 0009 5 5 18r3 r1+(2 5)r3 r2+3 5r3 r4+9 5r3 0100 0010 0001 0000 (3)?1C?z1/? 311 113 024 214 1 3r1 r2+(1)r1 r4+(2)r1 1 1 3 1 3 04 3 8 3 024 05 3 10 3 3 4r2 r1+(1 3)r2 r3+(2)r2 r4+5 3r2 101 012 000 000 2. (1) T5|?X?1C,? 9 111 314 123 r2+(3)r1 r3+(1)r1 111 027 034 1 2r2 r1+(1)r2 r3+3r2 10 5 2 017 2 0013 2 2 13r3 r1+(5 2)r3 r2+7 2r3 100 010 001 l?du x1= 0 x2= 0 x3= 0 =?) (2) T5|?X?1C,? 1523 2421 5361 r2+(2)r1 r3+(5)r1 1523 01427 028414 1 14r2 r1+5r2 r3+(28)r2 10 9 7 1 2 011 7 1 2 0000 l?du ( x1+ 9 7x3 1 2x4 = 0 x2 1 7x3 + 1 2x4 = 0 ,?) x1= 9 7C1 + 1 2C2 x2= 1 7C1 1 2C2 x3= C1 x4= C2 , C1,C2? (3) T5|?X?1C,? 10 1231 2401 1295 1232 r2+(2)r1 r3+(1)r1 r4+r1 1231 0063 00126 0063 1 6r2 r1+(3)r2 r3+12r2 r4+(6)r2 1201 2 001 1 2 0000 0000 l?du ( x1+ 2x2 1 2x4 = 0 x3+ 1 2x4 = 0 ,?) x1= 2C1+ 1 2C2 x2= C1 x3= 1 2C2 x4= C2 , C1,C2? 3.(1) T5|?O2?1C 1245 2314 38213 r2+(2)r1 r3+(3)r1 1245 07714 0141428 1 7r2 r1+2r2 r3+(14)r2 1021 0112 0000 l?du ( x1+ 2x3= 1 x2 x3= 2 -x3= c,?) x1= 1 2c x2= 2 + c x3= c , c? (2) T5|?O2?1C 11 23010 31542 071054 361591 1 2r1 r2+(3)r1 r4+(3)r1 1 3 2 01 2 0 07 2 55 2 2 071054 021 2 1515 2 1 2 7r2 r1+(3 2)r2 r3+(7)r2 r4+21 2 r2 10 15 7 11 7 6 7 0110 7 5 7 4 7 00000 00005 r3r4 10 15 7 11 7 6 7 0110 7 5 7 4 7 00005 00000 1 5r3 r1+(6 7)r3 r2+4 7r3 10 15 7 11 7 0 0110 7 5 7 0 00001 00000 l?du x1+ 15 7 x3 11 7 x4= 0 x2 10 7 x3+ 5 7x4 = 0 0 = 1 , T|). (3)T5|?O2?1C 1210 3214 1116 r2+(3)r1 r3+(1)r1 1210 0844 0306 1 8r2 r1+(2)r2 r3+3r2 1001 011 2 1 2 003 2 9 2 2 3r3 r2+1 2r3 1001 0102 0013 l?du x1= 1 x2= 2 x3= 3 , =?). (4)T5|?O2?1C 12 11211 22101 11211 11112 r2+(2)r1 r3+(1)r1 r4+(1)r1 11211 00321 02402 00121 r2r3 11211 02402 00321 00121 1 2r2 r1+r2 10010 01201 00321 00121 1 3r3 r2+2r3 r4+r3 10010 0104 3 1 3 0012 3 1 3 000 4 3 4 3 3 4r4 r1+r4 r2+4 3r4 r3+2 3r4 10001 01001 00101 00011 l?du x1= 1 x2= 1 x3= 1 x4= 1 , =?). 4. T5|?X?1C,? 11 11 11 r2+(1)r1 r3+()r1 11 0 11 01 1 2 1 1r2 r1+(1)r2 r3(1)r2 10 + 1 011 002 + 2 6= 2 6= 1k“), = 2 = 1k“); ? = 2, 112 121 211 101 011 000 , “) x1= c x2= c x3= c , c?. 13 ? = 1, 111 111 111 111 000 000 , “) x1= c1 c2 x2= c1 x3= c2 , c1,c2?. 5. T5|?O2?1C 11230 21641 32p71 1161t r2+(2)r1 r3+(3)r1 r4+(1)r1 11230 01221 016 + p21 0244t (1)r2 r1+(1)r2 r3+r2 r4+2r2 10411 01221 008 + p00 0000t + 2 ?t 6= 2,|) ?t = 2,k) ?t = 2,p 6= 8,?du x1 4x3+ x4= 1 x2+ 2x3+ 2x4= 1 (8 + p)x3= 0 ,?) x1= 1 c x2= 1 2c x3= 0 x4= c c?. ?t = 2,p = 8,?du ( x1 4x3+ x4= 1 x2+ 2x3+ 2x4= 1 14 ,?) x1= 1 + 4c1 c2 x2= 1 2c1 2c2 x3= c1 x4= c2 c1,c2? SK1-4 1.?A?11?13?B?u3?Am Q1= 001 010 100 , rB?11?“k?B?12?u3?Bm Q2= 1k0 010 001 . d Q = Q1Q2= 001 010 100 1k0 010 001 = 001 010 1k0 2. 1210 1301 ! r2+(1)r1 1210 0111 ! r1+(2)r2 1032 0111 ! (1) P1= 32 11 ! 15 (2) P1AP = 32 11 ! 52 30 ! 12 13 ! = 96 22 ! 12 13 ! = 30 02 ! (3)?B = P1AP,KA = PBP1 A10= (PBP1)10 = PB10P1 = 12 13 ! 3100 0210 ! 32 11 ! = 310211 3103 210 ! 32 11 ! = 311 211211 2 310 311 3 2103 210 2 310 ! 3.(1) 111100 111010 111001 r2+(1)r1 r3+r1 111100 022110 020101 (1 2)r2 r1+(1)r2 r3+(2)r2 100 1 2 1 2 0 011 1 2 1 2 0 002011 (1 2)r3 r2+r3 100 1 2 1 2 0 010 1 2 0 1 2 0010 1 2 1 2 _? 1 2 1 2 0 1 2 0 1 2 0 1 2 1 2 (2) 16 221100 343010 123001 1 2r1 r2+(3)r1 r3+(1)r1 11 1 2 1 2 00 01 3 2 3 2 10 01 5 2 1 2 01 r1+(1)r2 r3+(1)r2 101210 01 3 2 3 2 10 001111 r1+r3 r2+(3 2)r3 100321 0103 5 2 3 2 001111 _? 321 3 5 2 3 2 111 (3) 110100 121010 223001 r2+r1 r3+2r1 110100 011110 043201 r1+r2 r3+(4)r2 101210 011110 001641 (1)r3 r1+(1)r3 r2+(1)r3 100431 010531 001641 _? 431 531 641 (4) 17 13161000 21000100 32000010 57180001 r2+(2)r1 r3+(3)r1 r4+(5)r1 13161000 052122100 073183010 084225001 (1 5)r2 r1+(3)r2 r3+7r2 r4+8r2 101 5 6 5 1 5 3 5 00 01 2 5 12 5 2 5 1 5 00 001 5 6 5 1 5 7 5 10 004 5 14 5 9 5 8 5 01 (5)r3 r1+5r3 r2+(2 5)r3 r4+4 5r3 10000210 01000320 00161750 00021441 1 2r4 r3+(6)r4 10000210 01000320 00104573 00011 2 22 1 2 _? 0210 0320 4573 1 2 22 1 2 4. (1) 111100 254010 245001 r2+(2)r1 r3+(2)r1 111100 032210 023201 1 3r2 r1+(1)r2 r3+(2)r2 101 3 5 3 1 3 0 012 3 2 3 1 3 0 005 3 2 3 2 3 1 3 5r3 r1+1 3r3 r2+2 3r3 100 9 5 1 5 1 5 0102 5 3 5 2 5 000 2 5 2 5 3 5 d_? 9 5 1 5 1 5 2 5 3 5 2 5 2 5 2 5 3 5 18 X = 9 5 1 5 1 5 2 5 3 5 2 5 2 5 2 5 3 5 03 48 19 = 12 20 11 (2) 111100 210010 211001 r2+(2)r1 r3+(2)r1 111100 032210 033201 1 3r2 r1+r2 r3+(3)r2 10 1 3 1 3 1 3 0 012 3 2 3 1 3 0 001011 r3 r1+1 3r3 r2+2 3r3 100 1 3 0 1 3 0102 3 12 3 001011 d_? 1 3 0 1 3 2 3 12 3 011 X = 1 3 0 1 3 2 3 12 3 011 113 214 ! = 122 054 ! (3) 19 111100 111010 111001 1r1 r2+(1)r1 r3+(1)r1 111100 002110 020101 r2r3 111100 020101 002110 1 2r2 r1+r2 1011 2 0 1 2 010 1 2 0 1 2 002110 1 2r3 r1+r3 1000 1 2 1 2 010 1 2 0 1 2 001 1 2 1 2 0 d_? 0 1 2 1 2 1 2 0 1 2 1 2 1 2 0 X = 0 1 2 1 2 1 2 0 1 2 1 2 1 2 0 12 10 12 = 52 104 21 K ! WK 1. (A+B)(AB) = A2AB +BAB2,d(A+B)(AB) = A2B2?7 AB = BA. 2. 20 T E = 2 2 ! ? 23 ? 10 01 ! = 46 46 ! 10 01 ! = 36 45 ! 3. A = T = 1 2 1 ? 111 ? = 111 222 111 A A = 111 222 111 111 222 111 = 4 111 222 111 A8= 47 111 222 111 4. (A 2E) = 400 130 004 2 100 010 001 = 200 110 002 21 200100 110010 002001 1 2r1 r2+(1)r1 100 1 2 00 0101 2 10 002001 1 2r3 100 1 2 00 0101 2 10 00100 1 2 d (A 2E)1= 1 2 00 1 2 10 00 1 2 5. ?P?u?C 1n1-2?1?1, d P = 100 012 001 ?! JK 1.uA, ? A = 10 01 ! ,B = 00 10 ! ,C = 00 00 ! dB 6= C. uB, ? B = 10 01 ! ,C = 01 10 ! . uC,A,B. 2.(ABT)1= (BT)1A1 4. 11 210 ! 11 01 + 22 ! e5|) ? = 1 2. 5. A3 E = (A E)(A2+ A + E) = (A2+ A + E)(A E) = E 22 A3+ E = (A + E)(A2 A + E) = (A2 A + E)(A + E) = E ?A + EA E_?. n! )K 1. 1 2 00 0 1 3 0 00 1 3 1 = 200 030 003 111100 111010 111001 r2+(1)r1 r3+r1 111100 022110 002101 1 2r2 r1+r2 100 1 2 1 2 0 0111 2 1 2 0 002101 1 2r3 r2+r3 100 1 2 1 2 0 0100 1 2 1 2 001 1 2 0 1 2 d 111 111 111 1 = 1 2 1 2 0 0 1 2 1 2 1 2 0 1 2 X = 200 030 003 1 2 11 2 1 3 1 11 1 3 1 2 1 2 0 0 1 2 1 2 1 2 0 1 2 = 122 613 331 1 2 1 2 0 0 1 2 1 2 1 2 0 1 2 = 1 2 3 2 0 9 2 7 2 2 131 2.T5|?X?1C,? 23 1 + 11 11 + 1 111 + r3r1 111 + 11 + 1 1 + 11 r2+(1)r1 r3+(1)r1 111 + 0 02 2 1 r2 r1+(1)r2 r3+r2 102 + 011 002 3 6= 0 6= 3k“); = 0k“) 111 111 111 111 000 000 , “) x1= C1 C2 x2= C1 x3= C2 , C1,C2?. ? = 3k“), 211 121 112 101 011 000 , “) x1= C x2= C x3= C , C?. 3. 24 A A = 101 020 101 101 020 101 = 202 040 202 = 2 101 020 101 ? An= 2n1 101 020 101 d An 2An1= 2n1 101 020 101 2 2n2 101 020 101 = 0 o! yK Ak?1?n?,A A = nA. (E A)(E 1 n 1A) = E 1 n 1A A + 1 n 1A 2 = 1 n 1A 2 n n 1A + E = n n 1A n n 1A + E = E (E A)1= E 1 n 1A 25 1? SK2-1 1. (1) (634521) = 5 + 2 + 2 + 2 + 1 = 12 (2) (53142) = 4 + 2 + 1 = 7 (3) (123454321) = 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 (4) (135(2n1)(2n)(2n2)42) = 0+1+2+(n1)+(n1)+1+0 = n(n1) 2. D5= (1)(34512)a13a24a35a41a52= (1)6a13a24a35a41a52= a13a24a35a41a52 3. f(x) = P i1,i2,i3,i4S4(1) (i1,i2,i3,i4)a1i 1a2i2a3i3a4i4,?i46= 4deg(a1i1a2i2a3i3a4i4) 2 ?,?i26= 2,deg(a1i1a2i2a3i3a4i4) 2,deg(a1i1a2i2a3i3a4i4) 3 ka11a23a32a44= 6x4a21a23a31a44= 3x4.x3X3,x4X-6. 4. (1) ? ? ? ? ? ? ? ? 121 310 232 ? ? ? ? ? ? ? ? = 1 1 2 + 2 0 2 + 1 3 3 1 1 2 0 3 1 2 3 2 = 3 (2) ? ? ? ? ? ? ? ? 253 047 223 ? ? ? ? ? ? ? ? = 2 4 3 + 5 7 (2) + 3 0 (2) 3 4 (2) 7 (2) 2 3 5 0 = 24 70 + 24 + 28 = 6 (3) ? ? ? ? ? ? ? ? abc bca cab ? ? ? ? ? ? ? ? = a c b + b a c + c b a c c c a a a b b b = 3abc a3 b3 c3 26 (4) ? ? ? ? ? ? ? ? 111 2aa + b2b a2abb2 ? ? ? ? ? ? ? ? = (a + b)b2+ 2ba2+ 2aab (a + b)a2 2ab2 2bab = b3 3ab2+ 3a2b a3 = (b a)3 SK2-2 1. (1) ? ? ? ? ? ? ? ? 121 211 342 ? ? ? ? ? ? ? ? = 2 6 + 8 3 8 + 4 = 3 (2) ? ? ? ? ? ? ? ? ? ? ? 2143 42311 3092 1114 ? ? ? ? ? ? ? ? ? ? ? r2+(2)r1 = = = = = = = = r3+(3 2)r1 r4+(1 2r1 ? ? ? ? ? ? ? ? ? ? ? 2143 0055 03 2 35 2 0065 ? ? ? ? ? ? ? ? ? ? ? r2r3 = = = = = = ? ? ? ? ? ? ? ? ? ? ? 2143 03 2 35 2 0055 0065 ? ? ? ? ? ? ? ? ? ? ? r4+(6 5)r3 = = = = = = = = ? ? ? ? ? ? ? ? ? ? ? 2143 03 2 35 2 0055 0001 ? ? ? ? ? ? ? ? ? ? ? = 15 (3)11?-1?1 27 ? ? ? ? ? ? ? ? ? ? ? abbb aabb abab bbba ? ? ? ? ? ? ? ? ? ? ? = = ? ? ? ? ? ? ? ? ? ? ? abbb 0a b00 00a b0 b a00a b ? ? ? ? ? ? ? ? ? ? ? c1+c4 = = = = = ? ? ? ? ? ? ? ? ? ? ? a + bbbb 0a b00 00a b0 000a b ? ? ? ? ? ? ? ? ? ? ? = (a + b)(a b)3 (4)11?-1?1,2?1? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2aaa a2aa aa2a . . . . . . . . . . . aaa2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2aaa a 22 a00 a 202 a0 . . . . . . . . . . . a 2002 a ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? 2 + (n 1)aaaa 02 a00 002 a0 . . . . . . . . . . . 0002 a ? ? ? ? ? ? ? ? ? ? ? ? ? ? = (2 a)n12 + (n 1)a (5)11?-i?1i1,21i?i?1? 28 ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 + a111 22 + a22 333 + a3 . . . . . . . . . . . nnnn + a ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 + a111 2aa00 3a0a0 . . . . . . . . . . . na00a ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 + a + 2 + + n111 0a00 00a0 . . . . . . . . . . . 000a ? ? ? ? ? ? ? ? ? ? ? ? ? ? = a + (1 + n)n 2 an1 2. (1) 1?-1? ? ? ? ? ? ? ? ? a2(a + 1)2(a + 2)2 b2(b + 1)2(b + 2)2 c2(c + 1)2(c + 2)2 ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? a22a + 14a + 4 b22b + 14b + 4 c22c + 14c + 4 ? ? ? ? ? ? ? ? c3+(2)r2 = = = = = = = = ? ? ? ? ? ? ? ? a22a + 12 b22b + 12 c22c +