11X1 T01 10 matrices

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  • 1. Using Matrices to SolveSimultaneous Equations

2. Using Matrices to SolveSimultaneous Equations2 2 matrix a11 a12 A a21 a22 3. Using Matrices to SolveSimultaneous Equations2 2 matrix Determinant of A a11 a12 det A A a11a22 a21a12 A a21 a22 4. Using Matrices to SolveSimultaneous Equations2 2 matrix Determinant of A a11 a12 det A A a11a22 a21a12 A a21 a22 Inverse of A11 a22 a12 A A a21 a11 5. Using Matrices to SolveSimultaneous Equations2 2 matrix Determinant of A a11 a12 det A A a11a22 a21a12 A a21 a22 Inverse of A11 a22 a12 A A a21 a11 Solving simultaneousequations x mA y n x 1 m y A n 6. e.g. (i ) 2 x 3 y 21 (1)5x 2 y 3 (2) 7. e.g. (i ) 2 x 3 y 21 (1)5x 2 y 3 (2) 2 3 x 21 5 2 y 3 8. e.g. (i ) 2 x 3 y 21 (1)5x 2 y 3 (2) 2 3 x 21 5 2 y 3 x 1 2 3 21 y 11 5 2 3 9. e.g. (i ) 2 x 3 y 21 (1)5x 2 y 3 (2) A 2 2 5 3 2 3 x 21 5 2 y 3 x 1 2 3 21 y 11 5 2 3 10. e.g. (i ) 2 x 3 y 21 (1)5x 2 y 3 (2) A 2 2 5 3 2 3 x 21 5 2 y 3 x 1 2 3 21swap y 11 5 2 3 11. e.g. (i ) 2 x 3 y 21 (1)5x 2 y 3 (2) A 2 2 5 3 2 3 x 21 5 2 y 3 x 1 2 3 21swap y 11 5 2 3 change signs 12. e.g. (i ) 2 x 3 y 21 (1)5x 2 y 3 (2)A 2 2 5 3 2 3 x 21 5 2 y 3 x 1 2 3 21 swap y 11 5 2 3 x1 2 21 3 3 change y 11 5 21 2 3 signs 13. e.g. (i ) 2 x 3 y 21 (1)5x 2 y 3 (2)A 2 2 5 3 2 3 x 21 5 2 y 3 x 1 2 3 21 swap y 11 5 2 3 x1 2 21 3 3 change y 11 5 21 2 3 signs Multiply the row of the matrix with the columnof the vector 14. e.g. (i ) 2 x 3 y 21 (1)5x 2 y 3 (2)A 2 2 5 3 2 3 x 21 5 2 y 3 x 1 2 3 21 swap y 11 5 2 3 x1 2 21 3 3 change y 11 5 21 2 3 signs x1 33 Multiply the row of the y 11 99 matrix with the columnof the vector 15. e.g. (i ) 2 x 3 y 21 (1)5x 2 y 3 (2)A 2 2 5 3 2 3 x 21 5 2 y 3 x 1 2 3 21 swap y 11 5 2 3 x1 2 21 3 3 change y 11 5 21 2 3 signs x1 33 Multiply the row of the y 11 99 matrix with the columnof the vector x 3 y 9 16. e.g. (i ) 2 x 3 y 21 (1)5x 2 y 3 (2)A 2 2 5 3 2 3 x 21 5 2 y 3 x 1 2 3 21 swap y 11 5 2 3 x1 2 21 3 3 change y 11 5 21 2 3 signs x1 33 Multiply the row of the y 11 99 matrix with the columnof the vector x 3 y 9 x 3, y 9 17. (ii ) 171x 213 y 642 (1) 114 x 326 y 244(2) 18. (ii ) 171x 213 y 642 (1) 114 x 326 y 244(2) 171 213 x 642 114 326 y 244 19. (ii ) 171x 213 y 642 (1) 114 x 326 y 244(2) 171 213 x 642 114 326 y 244 x 1 326 213 642 y 31464 114 171 244 20. (ii ) 171x 213 y 642 (1) 114 x 326 y 244(2) 171 213 x 642 114 326 y 244 x 1 326 213 642 y 31464 114 171 244 x 1 157320 y 31464 31464 21. (ii ) 171x 213 y 642 (1) 114 x 326 y 244(2) 171 213 x 642 114 326 y 244 x 1 326 213 642 y 31464 114 171 244 x1 157320 y 31464 31464 x 5 y 1 22. (ii ) 171x 213 y 642 (1) 114 x 326 y 244(2) 171 213 x 642 114 326 y 244 x 1 326 213 642 y 31464 114 171 244 x1 157320 y 31464 31464 x 5 y 1 x 5, y 1 23. n n matrix a11 a1n A aann n1 24. n n matrix n A 1 k 1 a11 a1n a1k M 1kA k 1 where M1k is the matrix withaann n1 column 1 and row k removed 25. n n matrix n A 1 k 1 a11 a1n a1k M 1kA k 1 where M1k is the matrix withaann n1 column 1 and row k removed 3 2 1e.g. (i ) 1 4 2 0 1 2 26. n n matrixnA 1 k 1 a11 a1n a1k M 1kA k 1 where M1k is the matrix withaann n1column 1 and row k removed 3 2 1 4 21 21 4e.g. (i ) 1 4 2 3 2 1 20 20 1 0 1 2 27. n n matrix nA 1k 1 a11 a1n a1k M 1kA k 1 where M1k is the matrix withaann n1column 1 and row k removed 3 2 1 4 2 1 21 4e.g. (i ) 1 4 2 32 1 2 0 20 1 0 1 2 3 6 2 2 1 15 28. 1 3 5 20 1 3 4(ii )2 1 9 63 2 4 8 29. 1 3 5 2 1 3 4 0 3 4 0 1 40 1 30 1 3 4(ii ) 1 9 6 3 2 9 6 5 2 1 6 2 2 1 92 1 9 62 4 8 3 4 8 3 2 8 32 43 2 4 8 30. 1 3 5 21 3 40 3 4 0 1 4 0 1 30 1 3 4(ii ) 1 9 6 3 2 9 6 5 21 6 2 21 92 1 9 62 4 8 3 4 8 32 8 3 2 43 2 4 8 9 6161 9 96 2 6 2 9 34 3 0 4 3 4 4 8282 4 83 83 4 1 6 262 1 19 2 92 15 0 4 2 0 2 3 2 8 383 2 4 3 43 2 31. 1 3 5 2 1 3 40 3 40 1 40 1 30 1 3 4(ii ) 1 9 6 3 2 9 6 5 21 6 2 2 192 1 9 62 4 83 4 832 83 243 2 4 8 9 61 61 9 96 2 6 2 9 3 4 3 0 4 3 4 4 82 82 4 83 83 4 1 6 2 62 1 19 2 92 15 0 4 2 0 2 3 2 8 3 83 2 4 3 43 2 4 3 4 4 11 3 0 3 2 4 19 5 0 2 4 1 2 0 19 3 1 32. 1 3 5 2 1 3 40 3 40 1 40 1 30 1 3 4(ii ) 1 9 6 3 2 9 6 5 21 6 2 2 192 1 9 62 4 83 4 832 83 243 2 4 8 9 61 61 9 96 2 6 2 9 3 4 3 0 4 3 4 4 82 82 4 83 83 4 1 6 2 62 1 19 2 92 15 0 4 2 0 2 3 2 8 3 83 2 4 3 43 2 4 3 4 4 11 3 0 3 2 4 19 5 0 2 4 1 2 0 19 3 1 160 33. or convert to a triangular matrix (create a triangle of 0s)3 2 1e.g. (i ) 1 4 2 0 1 2 34. or convert to a triangular matrix (create a triangle of 0s)3 2 132 110 51e.g. (i ) 1 4 20 row2 row1 3 33 0 1 20 1 2 35. or convert to a triangular matrix (create a triangle of 0s)3 2 132 110 51e.g. (i ) 1 4 20 row2 row1 3 33 0 1 20 1 2 32 1105 0 33 3 row3 row2310 002 36. or convert to a triangular matrix (create a triangle of 0s)3 2 132 110 51e.g. (i ) 1 4 20 row2 row1 3 33 0 1 20 1 2 32 110 5 0 3 33 row3 row2 3 10 0 0 210 3 3 multiply the diagonal 3 2 37. or convert to a triangular matrix (create a triangle of 0s)3 2 132 110 51e.g. (i ) 1 4 20 row2 row1 3 33 0 1 20 1 2 32 1105 0 33 3 row3 row2310 0 02 10 3 3 multiply the diagonal3 2 15 38. 1 3 5 20 1 3 4(ii )2 1 9 63 2 4 8 39. 1 3 5 2 1 3 520 1 3 40 134(ii ) 2 1 9 6 0 51 2 row3 2 row13 2 4 8 0 7 11 2 row4 3 row1 40. 1 3 5 2 1 3 520 1 3 40 134(ii ) 2 1 9 6 0 51 2 row3 2 row13 2 4 8 0 7 11 2 row4 3 row11 3 520 1340 0 16 18 row3 5 row20 0 32 26 row4 7 row2 41. 1 3 5 2 1 35 20 1 3 40 1 3 4(ii ) 2 1 9 6 0 51 2 row3 2 row13 2 4 8 0 7 11 2 row4 3 row11 3 520 1340 0 16 18 row3 5 row20 0 32 26 row4 7 row21 3 520 1340 0 16 180 0 010 row4 2 row3 42. 1 3 5 2 1 35 20 1 3 40 1 3 4(ii ) 2 1 9 6 0 51 2 row3 2 row13 2 4 8 0 7 11 2 row4 3 row11 3 520 1340 0 16 18 row3 5 row20 0 32 26 row4 7 row21 3 520 1340 0 16 180 0 010 row4 2 row3 160 43. Adjoint matrixt M 11(1)M 1n n 1 adjA (1) n n1 M(1) nn1 M nn n1 44. Adjoint matrix t M 11(1)M 1n n 1 adjA i.e signs alternate (1) n n1 M(1) nn1 n1 M nn 45. Adjoint matrix t M 11(1)M 1n n 1 adjA i.e signs alternate (1) n n1 M(1) nn1 n1 M nn t means transpose rows and columns 46. Adjoint matrixt M 11(1)M 1n n 1 adjA i.e signs alternate (1) n n1 M(1) nn1 n1 M nn t means transpose rows and columns1adjAA A 47. Adjoint matrixt M 11(1)M 1n n 1 adjA i.e signs alternate (1) n n1 M(1) nn1 n1 M nn t means transpose rows and columns1adjAA Ae.g x 2 y z 5 (1)2 x 3 y 4 z 28 (2)4 x 5 y 3z 10 (3) 48. 1 2 1 A 2 3 4 4 5 3 49. 1 2 1 A 2 3 4 4 5 3 1 2 1A 2 344 5 3 50. 1 2 1 A 2 3 4 4 5 3 1 2 1 1 2 1A 2 34 0 7 64 5 30 3 1 51. 1 2 1 A 2 3 4 4 5 3 1 2 1 1 2 1 1 2 1A 2 34 0 7 6 0 7 64 5 30 3 1 11 0 0 7 52. 1 2 1 A 2 3 4 4 5 3 1 2 1 1 2 1 1 2 1A 2 34 0 7 6 0 7 6 114 5 30 3 1 11 0 0 7 53. 1 2 1 A 2 3 4 4 5 3 1 2 11 2 1 1 2 1A 2 34 0 7 6 0 7 6 114 5 30 3 1 11 0 0 7 32 3 t42 4 5 3 4 34 5 21 1 11 2adjA 53 4 34 5 21 1 11 2 3 42 4 2 3 54. 1 2 1 A 2 3 4 4 5 3 1 2 11 2 1 1 2 1A 2 34 0 7 6 0 7 6 114 5 30 3 1 11 0 0 7 32 3 t42 4 5 3 4 34 5 11 1 5 21 1 11 2 adjA 22 1 6 53 4 34 5 22 3 7 21 1 11 2 3 42 4 2 3 55. 1 2 1 x 5 2 3 4 y 28 4 5 3 z 10 56. 1 2 1 x 5 2 3 4 y 28 4 5 3 z 10 x 11 1 5 5 y 1 22 1 6 28 11 z 22 3 7 10 57. 1 2 1 x 5 2 3 4 y 28 4 5 3 z 10 x 11 1 5 5 y 1 22 1 6 28 11 z 22 3 7 10 x 3 y 2 z 4 58. 1 2 1 x 5 2 3 4 y 28 4 5 3 z 10 x 11 1 5 5 y 1 22 1 6 28 11 z 22 3 7 10 x 3 y 2 z 4 x 3, y 2, z 4 59. 1 2 1 x 5 2 3 4 y 28 4 5 3 z 10 x 11 1 5 5 y 1 22 1 6 28 11 z 22 3 7 10 x 3 y 2 z 4 x 3, y 2, z 4Exercise 1H; 1, 2, 6