Find all complex values for a where there's no solution for the non homogeneous system { x + ( 1 − a ) y + z = 1 2 x − y + z = 3 3 x − a y + ( a − 1 ) z = 4 After moving to a matrix representation and reduction, I have: ( 1 1 − a 1 | 1 0 2 a − 3 − 1 | 1 0 0 a − 1 | 0 ) Now, I don't see an a that answering this request, and I double-checked my reduction process.

Question

Find all complex values for   a where there&#039;s no solution for the non homogeneous system      {                                          x            +            (            1            −            a            )            y            +            z            =            1                                                            2            x            −            y            +            z            =            3                                                            3            x            −            a            y            +            (            a            −            1            )            z            =            4                                  After moving to a matrix representation and reduction, I have:      (                              1                                      1            −            a                                    1                                  |                          1                                      0                                      2            a            −            3                                                −            1                                            |                          1                                      0                          0                                      a            −            1                                            |                          0                      )  Now, I don&#039;t see an   a that answering this request, and I double-checked my reduction process.

Cahokiavv · Accepted Answer

If you rref you get this  [  1     0     0               3      a      −      4              2      a      −      3             0     1     0         1          2      a      −      3             0     0     1                    0        ]So  2  a  −  3  ≠  0  a  ≠      3    2  Now you still need to check   a  =  1 because if your calculations are correct, to get a   1 in the last row you have to divide by   a  −  1. If   a  =  1 you are dividing by   0I checked   a  =  1 and it works. Putting   a  =  1 leads to the system  [  1     0     0     1    0     1     0  −  1    0     0     1     0     ]And this works by substituting   a  =  1  ,  x  =  1  ,  y  =  −  1  ,  z  =  0 in the OP.I got it to this  [  1     (  1  −  a  )        1     1    0     (  2  a  −  3  )        −  1        1    0           0     (  a  −  3  )        0        ]So even if   a  =  3 it will not produce &#039;no solution&#039; because of   0 in last entry of row   3 , we would then have   0      /    0 which leads to a system with a row of   0&#039;s

Find all complex values for a where there's no solution for the non homogeneous system

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