how to solve system of linear inequations involving only ≠ inequalitiesSpecifically, I need to know how to find numbers x 1 , x 2 , … , x r such that they satisfy all of these inequalities: { a 11 x 1 + … + a 1 r x r ≠ b 1 a 21 x + … + a 2 r x r ≠ b 2 a s 1 x + … + a s r x r ≠ b s Is there any general way to find these numbers?

Question

how to solve system of linear inequations involving only    ≠  inequalitiesSpecifically, I need to know how to find numbers       x    1    ,      x    2    ,  …  ,      x    r   such that they satisfy all of these inequalities:      {                                        a                      11                                x            1                    +          …          +                      a                          1              r                                            x            r                                              ≠                      b            1                                                            a                      21                    x          +          …          +                      a                          2              r                                            x            r                                              ≠                      b            2                                                            a                          s              1                                x          +          …          +                      a                          s              r                                            x            r                                              ≠                      b            s                                  Is there any general way to find these numbers?

Semaj Stark · Accepted Answer

Let  A  =      [                                        a                      11                    +                      a                      12                    +                          ⋯                          +                      a                          1              r                                                                        a                      21                    +                      a                      22                    +                          ⋯                          +                      a                          2              r                                                            ⋮                               ⋮                          ⋱                          ⋮                                                  a                          s              1                                +                      a                          s              2                                +                          ⋯                          +                      a                          s              r                                            ]  and  X  =      [                                        x            1                                                            x            2                                                ⋮                                                  x            s                                ]    ,    B  =      [                                        b            1                                                            b            2                                                ⋮                                                  b            s                                ]  and Hence the given condition is  A  X  ≠  BLet       A    L          −      1       is the left inverse of the matrix   A such that       A    L          −      1        A  =      I    r   Therefore                          A        X        ≠        B                                    ⟹                                      A          L                      −            1                          A        X        ≠                  A          L                      −            1                          B                                    ⟹                                                      (                          A          L                      −            1                          A                          )                X        ≠                  A          L                      −            1                          B                                    ⟹                            X        ≠                  A          L                      −            1                          B            In this way you can find out which values       x    i   can not take

how to solve system of linear inequations involving only ≠ inequalities Sp

Answered question

Answer & Explanation

New Questions in Algebra I