Контрольная по ВМатем
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51. |
A = 0 |
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1 , B = 4 |
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1 ; |
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2 1 −1 |
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3 1 2 |
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−1 0 |
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5 −2 |
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52. |
A = 3 |
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2 , |
B = −2 |
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4 ; |
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4 −1 −2 |
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−1 6 −2 |
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53. |
A = 4 |
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B = 6 |
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4 ; |
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3 2 3 |
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5 −2 −3 |
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54. A = −1 |
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−3 |
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3 −1 0 1 |
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2 −2 1 −3 |
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6 |
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1 2 −2 1 |
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0 1 |
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55. |
A = 0 |
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−1 |
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−3 |
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2 −3 −1 |
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−3 2 −1 |
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56. |
A = 1 |
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0 , B = 0 |
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−1 0 3 |
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−1 0 2 |
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0 −1 |
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3 1 0 |
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57. |
A = 1 |
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0 , B = −3 |
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1 ; |
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−1 0 3 |
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0 −2 1 |
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1 −2 0 4 |
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3 −1 |
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58. |
A = 0 |
−1 |
2 |
0 |
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B = |
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−2 ; |
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1 2 −1 |
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1 −2 |
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59. |
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B = |
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A = |
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−2 |
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−1 0 |
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7 3 |
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60. |
AT = (1 |
−2 |
2 |
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2), B = (3 |
−3 |
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2 |
−1) ; |
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61. |
A = |
3 2 1 0 |
B = |
1 0 1 0 |
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2 |
1 |
0 3 , |
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1 |
1 |
0 |
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3 0 |
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−1 |
0 1 2 |
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62. |
A = |
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−3 |
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−1 , B = 0 |
−1 |
1 ; |
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1 0 1 |
1 0 2 |
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63. |
A = 0 |
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0 , B = 1 |
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1 ; |
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1 0 1 |
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0 1 0 |
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0 1 0 3 |
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0 3 0 2 |
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64. |
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−1 |
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T |
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2 |
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A = |
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= 3 |
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0 −2 |
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−1 0 1 0 |
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−2 −1 0 0 |
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1 0 −1 |
4 0 −2 |
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65. |
A = 1 |
−2 |
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0 , B = 0 |
1 |
5 ; |
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3 −1 0 |
4 −1 2 |
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−2 −1 0 |
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1 0 |
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66. |
A = 0 |
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2 , B = −1 |
1 |
−1 ; |
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1 0 1 |
2 0 −1 |
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0 −3 6 |
3 1 0 |
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67. |
A = 1 |
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0 , B = −3 |
0 |
3 ; |
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−1 0 3 |
0 3 1 |
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7 |
−1 |
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1 2 −2 1 |
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0 1 |
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68. |
A = |
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1 |
−1 |
5 |
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−3 |
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69. |
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1 |
−1 |
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A = −3 |
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−2 |
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1 |
0 |
1 |
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1 |
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70. |
A = 0 |
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0 , B = 1 |
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1 ; |
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1 0 1 |
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0 1 0 |
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2 4 −1 |
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5 2 1 |
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71. |
A = 1 |
−4 |
0 , B = 0 |
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2 ; |
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−1 −2 3 |
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−1 0 2 |
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3 2 1 0 |
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1 0 1 0 |
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72. |
A = 2 |
1 |
0 |
3 |
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B = |
2 |
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1 |
1 0 ; |
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1 |
0 |
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2 |
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73. |
A = −1 |
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1 , |
B = 0 |
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1 |
0 ; |
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2 −1 0 |
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−1 2 2 |
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−1 4 0 −2 |
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−2 −1 −3 1 |
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74. |
A = |
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1 0 1 0 |
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−2 1 0 2 |
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−2 1 3 3 |
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1 −2 2 −3 |
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−1 2 0 −2 |
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1 4 2 1 |
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1 0 0 |
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−2 −2 1 |
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75. |
A = 1 |
1 |
1 , B = 0 |
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0 ; |
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2 1 0 |
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3 2 1 |
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−1 |
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2 0 |
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1 |
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2 3 |
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76. |
A = 0 |
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−1 , |
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B = −2 |
3 2 ; |
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1 −1 1 |
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0 −1 −1 |
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77. |
2 −1 4 |
0 |
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T |
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−1 −2 |
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A = |
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B |
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−1 3 0 |
−2 |
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0 |
3 3 |
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0 1 2 −1 |
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T |
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−12 0 1 2 |
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78. |
A = |
−2 |
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−10 |
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3 |
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B |
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−3 −3 . |
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−5 |
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−1 |
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79. |
A = |
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−2 |
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0 |
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−1 |
−2 , B = |
0 |
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3 0 |
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1 −2 4 9 |
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−2 |
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−4 |
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−5 |
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−1 0 −2 |
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80. |
A = |
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1 |
−2 |
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B = |
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−2 |
2 3 0 |
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1 −1 4 7 |
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0 2 −1 3 |
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2 −1 0 2 |
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81. |
A = |
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1 4 −2 6 |
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B = |
0 1 −3 1 |
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−1 0 0 2 |
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−2 0 0 1 |
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1 |
−2 0 −2 |
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2 1 0 0 |
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0 1 0 1 |
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−1 −1 0 1 |
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82. |
A = |
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1 0 1 0 |
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B = |
0 1 0 1 |
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0 1 0 1 |
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−1 0 −1 1 |
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1 0 1 0 |
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1 1 0 0 |
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1 |
2 3 |
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−2 3 0 |
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83. |
A = −2 |
3 |
2 , |
B = −1 |
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0 −1 −1 |
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2 −1 0 |
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4 −2 1 |
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−4 2 1 |
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84. |
A = 1 |
−2 |
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B = 1 |
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−2 |
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−1 5 2 |
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0 −1 2 |
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6
85. A = 1
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86. A = 0
21
1
87. A = 0
2−1
88. A = 3
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89. A = 4
3
−1
90. A =
3
1
91. A =
0
2
92. A = 1
−1
−8 2 |
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5 |
7 |
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4 |
0 , B = 6 |
3 4 ; |
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−4 2 |
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−2 −3 |
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0 2 1 |
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1 −4 −1 −2 |
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−1 0 1 |
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−1 2 −1 |
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1 0 2 |
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3 |
−1 −2 2 |
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1 −2 2 |
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1 |
0 −2 1 |
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4 |
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1 |
1 , B = 4 |
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1 ; |
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1 |
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3 |
1 |
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0 |
2 |
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1 |
5 |
−2 |
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−1 2 , B = −2 0 4 ; |
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−1 −2 |
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−1 6 −2 |
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1 3 |
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2 |
5 |
7 |
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3 2 , B = 6 3 4 ; |
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2 3 |
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5 −2 −3 |
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2 −3 0 |
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1 1 0 1 |
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−1 0 1 |
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2 −2 1 −3 |
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6 |
1 |
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2 −2 1 |
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0 1 |
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3 −1 5 |
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B = |
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0 1 |
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−3 2 |
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−3 −1 |
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−3 2 −1 |
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4 |
0 , B = 0 1 0 ; |
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0 3 |
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−1 0 2 |
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0 −1 0 |
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3 1 0 |
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93. |
A = 1 |
2 |
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0 , B = −3 |
0 |
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1 ; |
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−1 0 3 |
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0 −2 1 |
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1 −2 0 4 |
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3 −1 |
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94. |
A = 0 |
−1 |
2 |
0 |
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B = |
−1 |
−2 ; |
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1 2 −1 |
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1 −2 |
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95. |
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B = |
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4 |
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A = |
1 |
3 |
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0 |
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−2 |
−1 0 |
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7 3 |
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96. |
AT = (1 |
−2 |
2 |
0 |
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2), B = (3 |
−3 |
0 |
2 −1) ; |
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97. |
3 2 1 0 |
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1 0 1 0 |
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A = 2 |
1 |
0 3 , |
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1 |
1 |
0 |
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3 0 |
−1 |
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0 1 2 |
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98. |
A = |
2 |
−3 |
−1 , B = 0 |
−1 |
1 ; |
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1 0 1 |
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1 0 2 |
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1 |
0 |
1 |
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1 |
0 |
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99. |
A = 0 |
1 |
0 , B = 1 |
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1 ; |
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1 0 1 |
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0 1 0 |
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0 1 0 3 |
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0 3 0 2 |
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100. A = |
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3 |
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3 |
2 |
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1 −1 |
1 , |
B |
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−2 . |
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−1 0 1 0 |
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−2 −1 0 0 |
Задача 3. Даны векторы a, b, c . Проверить, что векторы a, b, c об-
разуют базис и найти координаты вектора d в этом базисе. Вычислить скалярное и векторное произведение векторов a , b .
46
Вариант |
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a |
b |
c |
d |
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1 |
(1,0,1) |
(3,4,5) |
(3,8,1) |
(4,0,2) |
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2 |
(-1,2,0) |
(-3,2,4) |
(0,0,1) |
(6,5,-4) |
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3 |
(6,0,-1) |
(2,4,1) |
(3,3,1) |
(-2,5,7) |
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4 |
(1,-3,2) |
(5,0,2) |
(-2,4,1) |
(8,1,3) |
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5 |
(2,1,1) |
(-3,-1,5) |
(2,4,3) |
(6,1,2) |
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6 |
(6,3,2) |
(7,4,-5) |
(4,2,2) |
(4,1,2) |
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7 |
(1,-1,1) |
(2,1,3) |
(1,-2,2) |
(6,8,1) |
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8 |
(3,0,4) |
(6,3,4) |
(1,2,3) |
(-3,4,-2) |
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9 |
(7,2,1) |
(-1,2,1) |
(4,3,0) |
(5,-3,7) |
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10 |
(2,5,6) |
(1,2,2) |
(1,0,1) |
(3,2,2) |
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11 |
(1,0,1) |
(3,4,5) |
(3,8,1) |
(5,2,7) |
|||||||||
12 |
(-1,2,0) |
(-3,2,4) |
(0,0,1) |
(4,3,-1) |
|||||||||
13 |
(6,0,-1) |
(2,4,1) |
(3,3,1) |
(3,9,2) |
|||||||||
14 |
(1,-3,2) |
(5,0,2) |
(-2,4,1) |
(2,-1,-3) |
|||||||||
15 |
(2,1,1) |
(-3,-1,5) |
(2,4,3) |
(3,-1,4) |
|||||||||
16 |
(6,3,2) |
(7,4,-5) |
(4,2,2) |
(5,1,7) |
|||||||||
17 |
(1,-1,1) |
(2,1,3) |
(1,-2,2) |
(2,-8,1) |
|||||||||
18 |
(3,0,4) |
(6,3,4) |
(1,2,3) |
(-1,5,-3) |
|||||||||
19 |
(7,2,1) |
(-1,2,1) |
(4,3,0) |
(2,-2,2) |
|||||||||
20 |
(2,5,6) |
(1,2,2) |
(1,0,1) |
(4,3,2) |
|||||||||
21 |
(1,0,1) |
(3,4,5) |
(3,8,1) |
(-2,4,1) |
|||||||||
22 |
(-1,2,0) |
(-3,2,4) |
(0,0,1) |
(6,3,-1) |
|||||||||
23 |
(6,0,1) |
(2,4,1) |
(3,3,1) |
(8,-5,1) |
|||||||||
24 |
(1,-3,2) |
(5,0,2) |
(-2,4,1) |
(6,-1,2) |
|||||||||
25 |
(2,1,1) |
(-3,-1,5) |
(2,4,3) |
(7,8,-4) |
|||||||||
26 |
(6,3,2) |
(7,4,-5) |
(4,2,2) |
(5,4,3) |
|||||||||
27 |
(1,-1,1) |
(2,1,3) |
(1,-2,2) |
(-1,5,1) |
|||||||||
28 |
(3,0,4) |
(6,3,4) |
(1,2,3) |
(3,-2,-2) |
|||||||||
29 |
(7,2,1) |
(-1,2,1) |
(4,3,0) |
(9,-1,1) |
|||||||||
30 |
(2,5,6) |
(1,2,2) |
(1,0,1) |
(4,2,0) |
|||||||||
31 |
(3,4,5) |
(1,0,1) |
(3,8,1) |
(4,0,2) |
|||||||||
32 |
(-3,2,4) |
(-1,2,0) |
(0,0,1) |
(6,5,-4) |
|||||||||
33 |
(2,4,1) |
(6,0,-1) |
(3,3,1) |
(-2,5,7) |
47
34 |
(5,0,2) |
(1,-3,2) |
(-2,4,1) |
(8,1,3) |
35 |
(-3,-1,5) |
(2,2,1) |
(2,4,3) |
(6,1,2) |
36 |
(7,4,-5) |
(6,3,2) |
(4,2,2) |
(4,1,2) |
37 |
(2,1,3) |
(1,-1,1) |
(1,-2,2) |
(6,8,1) |
38 |
(6,3,4) |
(3,0,4) |
(1,2,3) |
(-3,4,-2) |
39 |
(-1,2,1) |
(7,2,1) |
(4,3,0) |
(5,-3,7) |
40 |
(1,2,2) |
(2,5,6) |
(1,0,1) |
(3,3,2) |
41 |
(3,4,5) |
(1,0,1) |
(3,8,1) |
(5,2,7) |
42 |
(-3,2,4) |
(-1,2,0) |
(0,0,1) |
(4,3,-1) |
43 |
(2,4,1) |
(6,0,-1) |
(3,3,1) |
(3,9,2) |
44 |
(5,0,2) |
(1,-3,2) |
(-2,4,1) |
(2,-1,-3) |
45 |
(-3,-1,5) |
(2,1,1) |
(2,4,3) |
(3,-1,4) |
46 |
(7,4,-5) |
(6,3,2) |
(4,2,2) |
(5,1,7) |
47 |
(2,1,3) |
(1,-1,1) |
(1,-2,2) |
(2,-8,1) |
48 |
(6,3,4) |
(3,0,4) |
(1,2,3) |
(-1,5,-3) |
49 |
(-1,2,1) |
(7,2,1) |
(4,3,0) |
(2,-2,2) |
50 |
(1,2,2) |
(2,5,6) |
(1,0,1) |
(4,3,2) |
51 |
(3,4,5) |
(1,0,1) |
(3,8,1) |
(-2,4,1) |
52 |
(-3,2,4) |
(-1,2,0) |
(0,0,1) |
(6,3,-1) |
53 |
(2,4,1) |
(6,0,1) |
(3,3,1) |
(8,-5,1) |
54 |
(5,0,2) |
(1,-3,2) |
(-2,4,1) |
(6,-1,2) |
55 |
(-3,-1,5) |
(2,1,1) |
(2,4,3) |
(7,8,-4) |
56 |
(7,4,-5) |
(6,3,2) |
(4,2,2) |
(5,4,3) |
57 |
(2,1,3) |
(1,-1,1) |
(1,-2,2) |
(-1,5,1) |
58 |
(6,3,4) |
(3,0,4) |
(1,2,3) |
(3,-2,-2) |
59 |
(-1,2,1) |
(7,2,1) |
(4,3,0) |
(9,-1,1) |
60 |
(1,2,2) |
(2,5,6) |
(1,0,1) |
(4,2,0) |
61 |
(3,8,1) |
(4,0,2) |
(3,4,5) |
(1,0,1) |
62 |
(0,0,1) |
(6,5,-4) |
(-3,2,4) |
(-1,2,0) |
63 |
(3,3,1) |
(-2,5,7) |
(2,4,1) |
(6,0,-1) |
64 |
(-2,4,1) |
(8,1,3) |
(5,0,2) |
(1,-3,2) |
65 |
(2,4,3) |
(6,1,2) |
(-3,-1,5) |
(2,2,1) |
66 |
(4,2,2) |
(4,1,2) |
(7,4,-5) |
(6,3,2) |
67 |
(1,-2,2) |
(6,8,1) |
(2,1,3) |
(1,-1,1) |
68 |
(1,2,3) |
(-3,4,-2) |
(6,3,4) |
(3,0,4) |
48
69 |
(4,3,0) |
(5,-3,7) |
(-1,2,1) |
(7,2,1) |
70 |
(1,0,1) |
(3,3,2) |
(1,2,2) |
(2,5,6) |
71 |
(3,8,1) |
(5,2,7) |
(3,4,5) |
(1,0,1) |
72 |
(0,0,1) |
(4,3,-1) |
(-3,2,4) |
(-1,2,0) |
73 |
(3,3,1) |
(3,9,2) |
(2,4,1) |
(6,0,-1) |
74 |
(-2,4,1) |
(2,-1,-3) |
(5,0,2) |
(1,-3,2) |
75 |
(2,4,3) |
(3,-1,4) |
(-3,-1,5) |
(2,1,1) |
76 |
(4,2,2) |
(5,1,7) |
(7,4,-5) |
(6,3,2) |
78 |
(1,-2,2) |
(2,-8,1) |
(2,1,3) |
(1,-1,1) |
79 |
(1,2,3) |
(-1,5,-3) |
(6,3,4) |
(3,0,4) |
80 |
(4,3,0) |
(2,-2,2) |
(-1,2,1) |
(7,2,1) |
81 |
(1,0,1) |
(4,3,2) |
(1,2,2) |
(2,5,6) |
82 |
(3,8,1) |
(-2,4,1) |
(3,4,5) |
(1,0,1) |
83 |
(0,0,1) |
(6,3,-1) |
(-3,2,4) |
(-1,2,0) |
84 |
(3,3,1) |
(8,-5,1) |
(2,4,1) |
(6,0,1) |
85 |
(-2,4,1) |
(6,-1,2) |
(5,0,2) |
(1,-3,2) |
86 |
(2,4,3) |
(7,8,-4) |
(-3,-1,5) |
(2,1,1) |
87 |
(4,2,2) |
(5,4,3) |
(7,4,-5) |
(6,3,2) |
88 |
(1,-2,2) |
(-1,5,1) |
(2,1,3) |
(1,-1,1) |
89 |
(1,2,3) |
(3,-2,-2) |
(6,3,4) |
(3,0,4) |
90 |
(4,3,0) |
(9,-1,1) |
(-1,2,1) |
(7,2,1) |
91 |
(3,8,1) |
(5,2,7) |
(1,0,1) |
(3,4,5) |
92 |
(0,0,1) |
(4,3,-1) |
(-1,2,0) |
(-3,2,4) |
93 |
(3,3,1) |
(3,9,2) |
(6,0,-1) |
(2,4,1) |
94 |
(-2,4,1) |
(2,-1,-3) |
(1,-3,2) |
(5,0,2) |
95 |
(2,4,3) |
(3,-1,4) |
(2,2,1) |
(-3,-1,5) |
96 |
(4,2,2) |
(5,1,7) |
(6,3,2) |
(7,4,-5) |
97 |
(1,-2,2) |
(2,-8,1) |
(1,-1,1) |
(2,1,3) |
98 |
(1,2,3) |
(-1,5,-3) |
(3,0,4) |
(6,3,4) |
99 |
(4,3,0) |
(2,-2,2) |
(7,2,1) |
(-1,2,1) |
100 |
(3,8,1) |
(5,2,7) |
(2,5,6) |
(3,4,5) |
Задача 4. Решить систему методом Крамера и матричным методом.
49
x + x + x = −2
1 2 3
1.2x1 − 3x2 − x3 = −6
3x1 + 4x2 + 3x3 = −5
3x + x + x = 8
1 2 3
3.2x1 − 3x2 + 2x3 = 2
x1 + 2x2 − x3 = 2
x + 2x − 4x = −7
1 2 3
2.2x1 − 3x2 + 5x3 = 113x1 − x2 + 5x3 = 16
2x + 3x − x = 4
|
1 |
2 |
3 |
4. x1 + x2 + 3x3 = 5 |
|||
|
|
− 4x2 + 3x3 = 5 |
|
3x1 |
|||
x |
+ 2x + 3x = 1 |
||
1 |
|
2 |
3 |
5. 5x1 + 8x2 − x3 = 7 |
|||
|
|
− 3x2 + 2x3 = 9 |
|
2x1 |
|||
3x + 4x + 2x = 8 |
|||
|
1 |
2 |
3 |
7. x1 |
+ 5x2 + 2x3 = 5 |
2x1 + 3x2 + 4x3 = 3
2x − x + 5x = 4
1 2 3
9.5x1 + 2x2 + 13x3 = 23x1 − x2 + 5x3 = 0
x + 3x = −6
2 3
11.x1 − 2x2 − x3 = 5
3x1 + 4x2 − 2x3 = 13
x + 2x − x = 2
1 2 3
13.2x1 − 3x2 + 2x3 = 2
3x1 + x2 + x3 = 8
x + 2x + 3x = 6
1 2 3
15.2x1 + 3x2 − x3 = 43x1 + x2 − 4x3 = 0
2x + 4x + x = 4
1 2 3
17.3x1 + 6x2 + 2x3 = 44x1 − x2 − 3x3 = 1
|
x − x |
− 3x |
= 13 |
|
|
1 |
2 |
3 |
|
19. |
2x1 + x2 − x3 |
= 0 |
||
|
|
− 2x2 + 4x3 = −15 |
||
|
3x1 |
|||
|
x + 2x |
+ x = 1 |
||
|
1 |
2 |
3 |
|
6. |
x2 − x3 = −3 |
|
||
|
|
|
|
|
|
4x1 + 2x2 − x3 = 0 |
x − x − x = 0
1 2 3
8.x1 + 2x2 + x3 = 1
x2 − x3 = −3
x + 5x + x = 0
1 2 3
10.2x1 − 4x2 − 3x3 = −13x1 + 4x2 + 2x3 = 8
4x1 + 2x2 − x3 = 0
12. x1 + 2x2 + x3 = 8
x2 − x3 = −3
50