Solve the following system of linear equations in terms of parameter a ∈ R and explain geometric interpretation of this system: a x + y + z = 1 , 2 x + 2 a y + 2 z = 3 , x + y + a z = 1.By Cronecker Capelli's theorem, we get: [ a 1 1 1 2 2 a 2 3 1 1 a 1 ] Row echelon form of this matrix is [ 1 1 a 1 0 2 ( a − 1 ) 2 ( 1 − a ) 1 0 0 ( 1 − a ) ( a + 2 ) ( 3 − 2 a ) / 2 ] System is inconsistent for a = 1 ∨ a = − 2. For every other value of a, system has unique solution.For every a instead of a = 1 ∧ a = − 2 there are three planes that intersect at a point.Question: Is this geometric interpretation correct?

Question

Solve the following system of linear equations in terms of parameter   a  ∈      R   and explain geometric interpretation of this system:  a  x  +  y  +  z  =  1  ,  2  x  +  2  a  y  +  2  z  =  3  ,  x  +  y  +  a  z  =  1.By Cronecker Capelli&#039;s theorem, we get:      [                            a                          1                          1                          1                                      2                          2          a                          2                          3                                      1                          1                          a                          1                      ]  Row echelon form of this matrix is      [                            1                          1                          a                          1                                      0                          2          (          a          −          1          )                          2          (          1          −          a          )                          1                                      0                          0                          (          1          −          a          )          (          a          +          2          )                          (          3          −          2          a          )                      /                    2                      ]  System is inconsistent for   a  =  1  ∨  a  =  −  2. For every other value of   a, system has unique solution.For every   a instead of   a  =  1  ∧  a  =  −  2 there are three planes that intersect at a point.Question: Is this geometric interpretation correct?

Duncan Cox · Accepted Answer

When   a  ≠  1 and   a  ≠  −  2 the solution is  x  =  z  =            2      a      −      3              2      (      a      −      1      )      (      a      +      2      )        ,    y  =            3      a      −      1              2      (      a      −      1      )      (      a      +      2      )      When   a  =  1 the first and third equations are the same and inconsistent with the second one, i.e. the plane described by the first and third equations is parallel to the plane defined by the second equation. The augmented matrix has rank 2 while the coefficient matrix has rank 1.When   a  =  −  2 the augmented matrix has rank 3 and the coefficient matrix has rank 2, so again the three equations are inconsistent.

Solve the following system of linear equations in terms of parameter a ∈

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