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I got this linear algebra question I hope You Guys can assist me with

Below there is a inhomogeneous system of linear equations which I solve:

[itex]x_{1} + 2x_{2} + x_{3} = a [/itex]

[itex]3x_{1} + 4x_{2} + 2x_{3} = a - 3 [/itex]

[itex]-4x_{1} + 2x_{2} + x_{3} = 3 [/itex]

I then end up the following matrix:

1 0 0 0 -3a/5

0 1 1/2 (2a+3)/5

0 0 0 (-11a/10)

I´m then supose to prove if its possible to solve the system for every a-value.

By solving the equation (2a+3)/5) = 0. I then get an a-value a = -1/2 .

If I insert this a into the above matrix and row-reduce that matrix I get:

1 0 0 0

0 1 1/2 0

0 0 0 1

I then do some tests and then conclude if I choose an a-value in the interval

[-1000000,1000000]. I still end up the same matrix above.

Is it then correct to assume if I chose an a-value in the interval [-infty,infty]. I would then still end up with the same matrix?

If my assumption is correct, is it then safe to assume that its possible to solve the system of linear-equations for every a-value??

sincerely

Fred