Determine the value of a if the system x 1 + 4 x 2 − 3 x 3 + 2 x 4 = 0 2 x 1 + 7 x 2 − 4 x 3 + 4 x 4 = 0 − x 1 + a x 2 + 5 x 3 − 2 x 4 = 0 3 x 1 + 10 x 2 − 5 x 3 + ( a ² + 4 a + 1 ) x 4 = 0 has more then 1 solution

Question

Determine the value of a if the system                              x          1                +        4                  x          2                −        3                  x          3                +        2                  x          4                                    =        0                            2                  x          1                +        7                  x          2                −        4                  x          3                +        4                  x          4                                    =        0                            −                  x          1                +        a                  x          2                +        5                  x          3                −        2                  x          4                                    =        0                            3                  x          1                +        10                  x          2                −        5                  x          3                +        (        a              ²        +        4        a        +        1        )                  x          4                                    =        0            has more then 1 solution

Tanner Hamilton · Accepted Answer

We know that a homogeneous linear system has more than one solution if and only if the rank of the matrix of the system is less than the number of unknowns. See Rank-nullity theorem.One of the possibilities how to find rank is to use row operations. You could start like this:      (                            1                          4                          −          3                          2                                      2                          7                          −          4                          4                                      −          1                          a                          5                          −          2                                      3                          10                          −          5                                      a            2                    +          4          a          +          1                      )    ∼      (                            1                          4                          −          3                          2                                      0                          −          1                          2                          0                                      0                          a          +          4                          2                          0                                      3                          10                          −          5                                      a            2                    +          4          a          +          1                      )    ∼      (                            1                          4                          −          3                          2                                      0                          1                          −          2                          0                                      0                          a          +          4                          2                          0                                      3                          10                          −          5                                      a            2                    +          4          a          +          1                      )    ∼      (                            1                          4                          −          3                          2                                      0                          1                          −          2                          0                                      0                          a          +          5                          0                          0                                      3                          10                          −          5                                      a            2                    +          4          a          +          1                      )  If   a  +  5  =  0, i.e. if   a  =  5, then you have a matrix which does not have full rank. So in this case you have that there is more than one solution.If   a  +  5  ≠  0, you can divide the third row by   (  a  +  5  )      (                            1                          4                          −          3                          2                                      0                          1                          −          2                          0                                      0                          a          +          5                          0                          0                                      3                          10                          −          5                                      a            2                    +          4          a          +          1                      )    ∼      (                            1                          4                          −          3                          2                                      0                          1                          −          2                          0                                      0                          1                          0                          0                                      3                          10                          −          5                                      a            2                    +          4          a          +          1                      )    ∼      (                            1                          0                          −          3                          2                                      0                          0                          −          2                          0                                      0                          1                          0                          0                                      3                          0                          −          5                                      a            2                    +          4          a          +          1                      )    ∼      (                            1                          0                          −          3                          2                                      0                          1                          0                          0                                      0                          0                          1                          0                                      3                          0                          −          5                                      a            2                    +          4          a          +          1                      )    ∼      (                            1                          0                          0                          2                                      0                          1                          0                          0                                      0                          0                          1                          0                                      3                          0                          0                                      a            2                    +          4          a          +          1                      )

Determine the value of a if the system <mtable columnalign="right left right left right left ri

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