ejercicios sistemas lineales, Ejercicios de Álgebra Lineal

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Universidad de CaldasPrimer Taller de Algebra LinealTema: Sistemas de E ua iones LinealesCarrera: GeologaProf. Alvaro SalasTyped in TEX Manizales, abril 24 de 2008Resolver los siguientes sistemas :1. 8>>>>>>><>>>>>>>: 4x1 + 6x2 + 3x3 + 3x4 = 54x1 + 7x2 + 3x3 + 3x4 = 82x1 + 5x2 + 4x3 + 3x4 = 175x1 + 8x2 + 4x3 + 4x4 = 6R. x1 = 2; x2 = 3; x3 = 15; x4 = 8:2. 8>>>>>>><>>>>>>>: 5x1 + 2x2 + 2x3 + 2x4 = 24x1 + 2x2 + 3x3 + 2x4 = 144x1 + 2x2 + 4x3 + 3x4 = 157x1 + 3x2 + 3x3 + 3x4 = 4R. x1 = 2; x2 = 7; x3 = 18; x4 = 17:3. 8>>>>>>>><>>>>>>>: 6x1 + 5x2 + 6x3 + 3x4 = 142x1 + 2x2 + 8x3 + 3x4 = 62x1 + 2x2 + 7x3 + 3x4 = 37x1 + 6x2 + 8x3 + 4x4 = 16R. x1 = 4; x2 = 10; x3 = 3; x4 = 2: 4. 8>>>>>>><>>>>>>>>: 4x1 4x2 + 4x3 + 3x4 = 146x1 + 4x2 + 3x3 + 3x4 = 143x1 6x2 + 4x3 + 3x4 = 126x1 3x2 + 5x3 + 4x4 = 18R. x1 = 2; x2 = 2; x3 = 12; x4 = 6:5. 8>>>>>>><>>>>>>>: 3x1 3x2 + 7x3 + 3x4 = 74x1 + 3x2 + 4x3 + 2x4 = 132x1 3x2 + 7x3 + 3x4 = 54x1 2x2 + 9x3 + 4x4 = 1R. x1 = 2; x2 = 5; x3 = 5; x4 = 7:6. 8>>>>>>>><>>>>>>>: 3x1 + 2x2 + 4x3 + 2x4 = 143x1 + 3x2 + 7x3 + 3x4 = 32x1 + 2x2 + 7x3 + 3x4 = 44x1 + 3x2 + 6x3 + 3x4 = 12R. x1 = 18; x2 = 17; x3 = 3; x4 = 9:1 7. 8>>>>>>><>>>>>>>: 5x1 + 4x2 + 4x3 + 2x4 = 17x1 + 7x2 + 6x3 + 3x4 = 73x1 + 3x2 + 7x3 + 3x4 = 207x1 + 6x2 + 6x3 + 3x4 = 1R. x1 = 1; x2 = 6; x3 = 1; x4 = 16:8. 8>>>>>>><>>>>>>>: 6x1 + 5x2 + 2x3 + 2x4 = 25x1 + 6x2 + 4x3 + 3x4 = 44x1 + 5x2 + 4x3 + 3x4 = 18x1 + 7x2 + 3x3 + 3x4 = 4R. x1 = 5; x2 = 8; x3 = 7; x4 = 3:9. 8>>>>>>><>>>>>>>: 7x1 + 7x2 + 3x3 + 3x4 = 53x1 + 3x2 + 3x3 + 2x4 = 85x1 + 4x2 + 4x3 + 3x4 = 69x1 + 9x2 + 4x3 + 4x4 = 7R. x1 = 5; x2 = 4; x3 = 3; x4 = 1:10. 8>>>>>>>><>>>>>>>: 7x1 + 6x2 + 3x3 + 3x4 = 145x1 5x2 + 5x3 + 3x4 = 163x1 2x2 + 3x3 + 2x4 = 119x1 + 8x2 + 4x3 + 4x4 = 19R. x1 = 1; x2 = 0; x3 = 0; x4 = 7: 11. 8>>>>>>>><>>>>>>>: 3x1 + 4x2 + 5x3 + 3x4 = 24x1 + 6x2 + 4x3 + 3x4 = 124x1 + 5x2 + 4x3 + 3x4 = 35x1 + 6x2 + 6x3 + 4x4 = 7R. x1 = 11; x2 = 15; x3 = 5; x4 = 18:12. 8>>>>>>><>>>>>>>: 2x1 5x2 + 5x3 + 2x4 = 85x1 2x2 + 7x3 + 3x4 = 73x1 3x2 + 7x3 + 3x4 = 14x1 3x2 + 7x3 + 3x4 = 5R. x1 = 4; x2 = 2; x3 = 4; x4 = 15:13. 8>>>>>>><>>>>>>>: 5x1 + 4x2 + 4x3 + 3x4 = 66x1 + 7x2 + 3x3 + 3x4 = 154x1 + 4x2 + 4x3 + 3x4 = 17x1 + 6x2 + 5x3 + 4x4 = 3R. x1 = 5; x2 = 6; x3 = 8; x4 = 9:14. 8>>>>>>>><>>>>>>>: 7x1 + 4x2 + 2x3 + 2x4 = 75x1 + 3x2 + 2x3 + 2x4 = 24x1 2x2 + 4x3 + 3x4 = 119x1 + 5x2 + 3x3 + 3x4 = 6R. x1 = 1; x2 = 3; x3 = 9; x4 = 15:2 31. 8>>>>>>><>>>>>>>: 5x1 9x2 + 5x3 + 3x4 = 34x1 8x2 + 5x3 + 3x4 = 63x1 4x2 + 3x3 + 2x4 = 17x1 8x2 + 6x3 + 4x4 = 1R. x1 = 1; x2 = 2; x3 = 16; x4 = 18:32. 8>>>>>>><>>>>>>>: 5x1 + 2x2 + 2x3 + 2x4 = 135x1 + 3x2 + 2x3 + 2x4 = 202x1 + 2x2 + 4x3 + 3x4 = 167x1 + 3x2 + 3x3 + 3x4 = 20R. x1 = 1; x2 = 7; x3 = 2; x4 = 4:33. 8>>>>>>><>>>>>>>: 3x1 + 2x2 + 6x3 + 2x4 = 145x1 + 6x2 + 7x3 + 3x4 = 123x1 + 3x2 + 5x3 + 2x4 = 125x1 + 4x2 + 8x3 + 3x4 = 11R. x1 = 16; x2 = 3; x3 = 5; x4 = 13:34. 8>>>>>>>><>>>>>>>: 4x1 + 5x2 + 3x3 + 3x4 = 132x1 5x2 + 4x3 + 3x4 = 42x1 6x2 + 4x3 + 3x4 = 65x1 + 6x2 + 4x3 + 4x4 = 16R. x1 = 0; x2 = 2; x3 = 3; x4 = 2: 35. 8>>>>>>><>>>>>>>: 6x1 + 5x2 + 2x3 + 2x4 = 83x1 2x2 + 3x3 + 2x4 = 112x1 3x2 + 3x3 + 2x4 = 168x1 + 7x2 + 3x3 + 3x4 = 7R. x1 = 5; x2 = 0; x3 = 4; x4 = 7:36. 8>>>>>>><>>>>>>>: 7x1 + 5x2 + 6x3 + 3x4 = 54x1 + 5x2 + 4x3 + 2x4 = 112x1 2x2 + 7x3 + 3x4 = 129x1 + 7x2 + 8x3 + 4x4 = 2R. x1 = 9; x2 = 5; x3 = 7; x4 = 3:37. 8>>>>>>><>>>>>>>: 5x1 + 2x2 + 4x3 + 2x4 = 52x1 4x2 + 5x3 + 2x4 = 202x1 5x2 + 5x3 + 2x4 = 177x1 + 3x2 + 6x3 + 3x4 = 3R. x1 = 9; x2 = 3; x3 = 16; x4 = 15:38. 8>>>>>>>><>>>>>>>: 5x1 + 4x2 + 4x3 + 2x4 = 22x1 4x2 + 5x3 + 2x4 = 112x1 5x2 + 5x3 + 2x4 = 197x1 + 5x2 + 6x3 + 3x4 = 1R. x1 = 12; x2 = 8; x3 = 15; x4 = 15:5 39. 8>>>>>>><>>>>>>>>: 5x1 + 4x2 + 2x3 + 2x4 = 133x1 + 2x2 + 3x3 + 2x4 = 54x1 + 3x2 + 4x3 + 3x4 = 87x1 + 6x2 + 3x3 + 3x4 = 12R. x1 = 15; x2 = 13; x3 = 14; x4 = 9:40. 8>>>>>>><>>>>>>>: 4x1 + 5x2 + 3x3 + 3x4 = 62x1 + 5x2 + 4x3 + 3x4 = 12x1 + 4x2 + 4x3 + 3x4 = 55x1 + 7x2 + 4x3 + 4x4 = 7R. x1 = 1; x2 = 4; x3 = 7; x4 = 17:41. 8>>>>>>><>>>>>>>: 3x1 + 2x2 + 6x3 + 2x4 = 12x1 + 2x2 + 8x3 + 3x4 = 32x1 + 2x2 + 7x3 + 3x4 = 74x1 + 3x2 + 8x3 + 3x4 = 2R. x1 = 7; x2 = 15; x3 = 4; x4 = 17: 42. 8>>>>>>><>>>>>>>>: 7x1 + 3x2 + 3x3 + 3x4 = 135x1 + 3x2 + 2x3 + 2x4 = 195x1 + 2x2 + 4x3 + 3x4 = 29x1 + 4x2 + 4x3 + 4x4 = 16R. x1 = 4; x2 = 9; x3 = 2; x4 = 16:43. 8>>>>>>><>>>>>>>>: 7x1 + 6x2 + 2x3 + 2x4 = 123x1 + 3x2 + 4x3 + 2x4 = 63x1 + 3x2 + 3x3 + 2x4 = 89x1 + 8x2 + 3x3 + 3x4 = 11R. x1 = 2; x2 = 10; x3 = 2; x4 = 19: Resolver los siguientes sistemas llevando la matriz ampliada a una forma es alonada por las. A partirde la solu ion general, en ontrar una solu ion parti ular y veri arla.44. 8>>>>>>><>>>>>>>: 11x1 + 9x2 + 8x3 + 2x4 + 13x5 = 410x1 + 9x2 + 8x3 + 2x4 + 10x5 = 26x1 + 7x2 + 9x3 + 2x4 + 10x5 = 615x1 + 13x2 + 12x3 + 3x4 + 17x5 = 5Resp. x1 = 2 3x5; x2 = 4x5 4; x3 = 4 4x5; x4 = 8x5 7:6 45. 8>>>>>>>><>>>>>>>: 19x1 + 14x2 + 3x3 + 2x4 + 6x5 = 124x1 5x2 + 5x3 + 2x4 + 16x5 = 202x1 6x2 + 5x3 + 2x4 + 11x5 = 1424x1 + 16x2 + 5x3 + 3x4 + 9x5 = 17Resp. x1 = 5x5 6; x2 = 5x5 + 6; x3 = 5x5 + 8; x4 = 2x5 3:46. 8>>>>>>><>>>>>>>: 14x1 + 12x2 + 5x3 + 3x4 + 4x5 = 1111x1 + 6x2 + 7x3 + 3x4 + 17x5 = 75x1 + 2x2 + 5x3 + 2x4 + 11x5 = 218x1 + 15x2 + 7x3 + 4x4 + 7x5 = 13Resp. x1 = 8 3x5; x2 = 3x5 9; x3 = 2x5 6; x4 = 4x5 + 5:47. 8>>>>>>><>>>>>>>>: 8x1 + 3x2 + 5x3 + 2x4 + 17x5 = 312x1 + 4x2 + 8x3 + 3x4 + 26x5 = 77x1 + 3x2 + 5x3 + 2x4 + 13x5 = 712x1 + 6x2 + 7x3 + 3x4 + 22x5 = 4Resp. x1 = 4x5 4; x2 = 3x5 + 2; x3 = 2x5 + 7; x4 = 2x5 3:48. 8>>>>>>><>>>>>>>: 7x1 + 9x2 + 12x3 + 3x4 + 11x5 = 37x1 + 9x2 + 13x3 + 3x4 + 14x5 = 55x1 + 8x2 + 13x3 + 3x4 + 9x5 = 19x1 + 12x2 + 16x3 + 4x4 + 13x5 = 3Resp. x1 = 3 5x5; x2 = 5x5 2; x3 = 2 3x5; x4 = 5x5 8: 7 57. 8>>>>>>>><>>>>>>>: 11x1 + 12x2 + 8x3 + 2x4 + 21x5 = 710x1 + 15x2 + 13x3 + 3x4 + 18x5 = 95x1 + 9x2 + 9x3 + 2x4 + 10x5 = 714x1 + 17x2 + 12x3 + 3x4 + 24x5 = 7Resp. x1 = 3 5x5; x2 = 5x5 4; x3 = 6 4x5; x4 = 3x5 13:58. 8>>>>>>><>>>>>>>: 9x1 + 8x2 + 9x3 + 2x4 + 2x5 = 124x1 + 4x2 + 11x3 + 2x4 + 11x5 = 175x1 + 6x2 + 16x3 + 3x4 + 10x5 = 1512x1 + 11x2 + 14x3 + 3x4 + 4x5 = 17Resp. x1 = 5x5 9; x2 = 7x5 + 11; x3 = 3x5 3; x4 = 7x5 + 4:59. 8>>>>>>><>>>>>>>>: 3x1 + x2 + 12x3 + 2x4 + 20x5 = 25x1 + 3x2 + 17x3 + 3x4 + 19x5 = 66x1 + 4x2 + 16x3 + 3x4 + 14x5 = 66x1 + 3x2 + 17x3 + 3x4 + 24x5 = 4Resp. x1 = 5x5 2; x2 = 7x5 + 4; x3 = 2 3x5; x4 = 12x5 10:60. 8>>>>>>><>>>>>>>: 7x1 + 11x2 + 8x3 + 2x4 + 12x5 = 1111x1 + 18x2 + 11x3 + 3x4 + 8x5 = 83x1 + 13x2 + 13x3 + 3x4 + 8x5 = 169x1 + 16x2 + 12x3 + 3x4 + 14x5 = 12Resp. x1 = 4x5 7; x2 = 4x5 + 12; x3 = 6x5 10; x4 = 10x5 7: 10 61. 8>>>>>>>><>>>>>>>: x1 10x2 + 8x3 + 2x4 + 31x5 = 188x1 + 13x2 + 5x3 + 2x4 + 8x5 = 163x1 2x2 + 10x3 + 3x4 + 20x5 = 73x1 8x2 + 11x3 + 3x4 + 36x5 = 18Resp. x1 = 5x5 14; x2 = 2x5 + 5; x3 = 5 4x5; x4 = 13x5 + 3:62. 8>>>>>>><>>>>>>>: x1 + x2 + 11x3 + 2x4 + 14x5 = 119x1 + 7x2 + 16x3 + 3x4 + 19x5 = 182x1 + 2x2 + 11x3 + 2x4 + 12x5 = 136x1 + 5x2 + 16x3 + 3x4 + 18x5 = 18Resp. x1 = 4 5x5; x2 = 7x5 6; x3 = 2x5 3; x4 = 3x5 + 12:63. 8>>>>>>><>>>>>>>: 11x1 + 13x2 + 7x3 + 3x4 + 21x5 + 19x6 = 18x1 + 9x2 + 4x3 + 2x4 + 10x5 + 6x6 = 53x1 + 7x2 + 5x3 + 2x4 + 10x5 + 8x6 = 514x1 + 17x2 + 9x3 + 4x4 + 25x5 + 21x6 = 4Resp. x1 = 3x5 5x6 5; x2 = 4x5 + 8x6 + 9; x3 = 7x5 11x6 7; x4 = 3x5 + 3x6 464. 8>>>>>>><>>>>>>>: 9x1 + 6x2 + 3x3 + 2x4 + 7x5 + 6x6 = 1110x1 + 7x2 + 4x3 + 2x4 + 13x5 + 11x6 = 156x1 + 5x2 + 5x3 + 2x4 + 13x5 + 10x6 = 313x1 + 9x2 + 5x3 + 3x4 + 12x5 + 10x6 = 13Resp. x1 = 3x5 3x6 11; x2 = 3x5 + 3x6 + 11; x3 = 6x5 5x6 4; x4 = 10x5 + 9x6 + 17 11 65. 8>>>>>>>><>>>>>>>: 16x1 + 10x2 + 3x3 + 2x4 + 2x5 + 6x6 = 914x1 + 7x2 + 6x3 + 3x4 + 8x5 + 16x6 = 1x1 3x2 + 5x3 + 2x4 + 8x5 + 11x6 = 1920x1 + 12x2 + 5x3 + 3x4 + 4x5 + 10x6 = 6Resp. x1 = 4x5 5x6 + 12; x2 = 6x5 + 6x6 14; x3 = 2x5 6x6 3; x4 = 2x5 + 16x6 1766. 8>>>>>>><>>>>>>>: 9x1 + 7x2 + 9x3 + 3x4 + 14x5 + 16x6 = 152x1 + 3x2 + 8x3 + 2x4 + 12x5 + 20x6 = 82x1 + 3x2 + 7x3 + 2x4 + 8x5 + 13x6 = 511x1 + 9x2 + 12x3 + 4x4 + 17x5 + 20x6 = 16Resp. x1 = 3x5 2x6 8; x2 = 4x5 + 2x6 + 12; x3 = 4x5 7x6 + 3; x4 = 7x5 + 17x6 1867. 8>>>>>>><>>>>>>>: 9x1 + 6x2 + 4x3 + 2x4 + 14x5 + 14x6 = 7 4x2 + 8x3 + 3x4 + 24x5 + 24x6 = 75x1 + 2x2 + 7x3 + 3x4 + 18x5 + 18x6 = 313x1 + 9x2 + 6x3 + 3x4 + 19x5 + 19x6 = 8Resp. x1 = 4x5 4x6 + 5; x2 = 5x5 + 5x6 6; x3 = 4x5 + 4x6 7; x4 = 12x5 12x6 + 1368. 8>>>>>>>><>>>>>>>: 11x1 + 7x2 + 7x3 + 2x4 + 19x5 + 19x6 = 1713x1 + 8x2 + 11x3 + 3x4 + 21x5 + 21x6 = 513x1 + 11x2 + 10x3 + 3x4 + 11x5 + 11x6 = 816x1 + 12x2 + 10x3 + 3x4 + 21x5 + 21x6 = 19Resp. x1 = 5x5 5x6 10; x2 = 5x5 + 5x6 + 3; x3 = 5x5 + 5x6 + 6; x4 = 17x5 17x6 + 1512 78. 8>>>>>>><>>>>>>>>: 6x1 + 3x2 + 8x3 + 3x4 + 19x5 + 27x6 = 95x1 + 3x2 + 5x3 + 2x4 + 10x5 + 15x6 = 156x1 + 4x2 + 7x3 + 3x4 + 10x5 + 15x6 = 209x1 + 5x2 + 10x3 + 4x4 + 21x5 + 31x6 = 14Resp. x1 = 3x5 5x6 2; x2 = 4x5 + 6x6 14; x3 = 5x5 6x6 + 15; x4 = 9x5 + 11x6 1979. 8>>>>>>>><>>>>>>>: 10x1 + 6x2 + 16x3 + 3x4 + 9x5 + 13x6 = 312x1 + 6x2 + 17x3 + 3x4 + 12x5 + 19x6 = 111x1 + 7x2 + 16x3 + 3x4 + 8x5 + 13x6 = 114x1 + 9x2 + 21x3 + 4x4 + 11x5 + 16x6 = 2Resp. x1 = 3x5 2x6 2; x2 = 4x5 + 2x6; x3 = 3x5 2x6 + 2; x4 = 17x5 + 9x6 380. 8>>>>>>><>>>>>>>: 19x1 + 14x2 + 3x3 + 2x4 + 5x5 + 8x6 = 412x1 + 7x2 + 4x3 + 2x4 + 6x5 + 10x6 = 132x1 2x2 + 5x3 + 2x4 + 6x5 + 11x6 = 2024x1 + 17x2 + 5x3 + 3x4 + 7x5 + 12x6 = 4Resp. x1 = 3x5 3x6 16; x2 = 4x5 + 4x6 + 19; x3 = 6x5 + 5x6 + 4; x4 = 11x5 11x6 + 1581. 8>>>>>>>><>>>>>>>: 4x1 + 5x3 + 2x4 + 11x5 + 18x6 = 135x1 + x2 + 5x3 + 2x4 + 11x5 + 15x6 = 13x1 + 5x3 + 2x4 + 8x5 + 13x6 = 208x1 + 2x2 + 7x3 + 3x4 + 14x5 + 22x6 = 9Resp. x1 = 3x5 5x6 + 7; x2 = 3x5 + 8x6 + 7; x3 = 5x5 + 2x6 1; x4 = 13x5 4x6 1815 82. 8>>>>>>>><>>>>>>>: 7x1 + 4x2 + 8x3 + 2x4 + 17x5 + 20x6 = 188x1 + 7x2 + 6x3 + 2x4 + 7x5 + 8x6 = 77x1 + 7x2 + 10x3 + 3x4 + 8x5 + 13x6 = 1210x1 + 7x2 + 11x3 + 3x4 + 19x5 + 23x6 = 17Resp. x1 = 3x5 2x6; x2 = 3x5 + 2x6 + 5; x3 = 2x5 4x6 5; x4 = 4x5 + 9x6 + 183. 8>>>>>>><>>>>>>>: 11x1 + 6x2 + 6x3 + 2x4 + 24x5 + 23x6 = 717x1 + 13x2 + 6x3 + 3x4 + 14x5 + 11x6 = 713x1 + 10x2 + 7x3 + 3x4 + 11x5 + 14x6 = 1416x1 + 10x2 + 8x3 + 3x4 + 27x5 + 26x6 = 6Resp. x1 = 4x5 3x6 + 11; x2 = 3x5 + 4x6 12; x3 = 4x5 3x6 13; x4 = 13x5 + 2x6 + 1884. Partiendo de la respuesta, hallar la Forma Es alonada Redu ida de las matri es ampliadas de lossistemas orrespondientes a los problemas 40 al 84.85. Veri ar que ada uno de los sistemas dados are e de solu ion ( esto es, ada uno de ellos esin ompatible o in onsistente). a.8>>>>>>><>>>>>>>: x1 + x2 + 3x3 2x4 + 3x5 = 12x1 + 2x2 + 4x3 x4 + 3x5 = 23x1 + 3x2 + 5x3 2x4 + 3x5 = 12x1 + 2x2 + 8x3 3x4 + 9x5 = 2 b.8>>>>>>><>>>>>>>: 10x1 + 23x2 + 17x3 + 44x4 = 2515x1 + 35x2 + 26x3 + 69x4 = 4025x1 + 57x2 + 42x3 + 108x4 = 6530x1 + 69x2 + 51x3 + 133x4 = 9516 86. Investigar el siguiente sistema lineal en x, y y z dependiendo del valor de  :8>>>>><>>>>>: x + y +  z = 12x + 3y +  z = 2 x 3y + 2z = 2 :Resp. Para  6= 52 y  6= 2 el sistema admite solu ion uni a. Si  = 52 el sistema no tienesolu ion. Finalmemte, para  = 2 el sistema admite in nidad de solu iones.87. Halle un valor de p de modo que todas las entradas de la FER de la matriz0BBB1 2 3 4 56 7 8 9 102 6 p 3 41CCCAsean numeros enteros. Resp. p = 9, p = 11.88. Des riba todas las posibles FER para matri es de tamano : 2 2, 2 3, 4 3, 3 2, 3 3.89. Resolver los siguientes sistemas lineales llevando la matriz ampliada a su forma es alonadaredu ida (FER). A partir de la solu ion general (representada en forma parametri a) halle unasolu ion parti ular y verifquela. a. 8>>>><>>>>>: x1 + 2x2 + 3x3 + 4x4 = 32x1 + 5x2 + 6x3 + 8x4 = 73x1 + 10x2 + 9x3 + 12x4 = 13 b. 8>>>>>>>>>>><>>>>>>>>>>>: x1 + 2x2 + 3x3 + x4 = 3x1 + 4x2 + 5x3 + 2x4 = 22x1 + 9x2 + 8x3 + 3x4 = 73x1 + 7x2 + 7x3 + 2x4 = 125x1 + 7x2 + 9x3 + 2x4 = 20Resp. x1 = 1 3t 4s, x2 = 1, x3 = t, x4 = s(s, t 2 R). Resp. x1 = t2 + 316 , x2 = 23, x3 = t2 76,x4 = t (t 2 R). 17