For what values of the constants a and b is the system consistent?

[Wen Qualto]

I can´t figure out how to solve this. The question is: For what values of a and b, which are constants, is this system consistent? The thing that i´ve done is that i made an augmented matrix and then reduced it via Gauss-Jordan, the only answer i get is that the system is consistent for all values of a and b, but that does not feel right. I really appriciate the help!

 

[Deleted member 4993]

View attachment 21409

I can´t figure out how to solve this. The question is: For what values of a and b, which are constants, is this system consistent? The thing that i´ve done is that i made an augmented matrix and then reduced it via Gauss-Jordan, the only answer i get is that the system is consistent for all values of a and b, but that does not feel right. I really appriciate the help!

Your "proposed" method is correct. If you want us check your work and figure out your mistake (if there is any), You would have to show us your work.

Please write back with a copy of your work.

 

[Wen Qualto]

Your "proposed" method is correct. If you want us check your work and figure out your mistake (if there is any), You would have to show us your work.

Please write back with a copy of your work.

Thank you for your quick response. This is my solution, and the only conclusion i can think of is that the system is consistent for all values of a and b. I hope you can understand my writing.

 

[ETK]

Your conclusion, which you could have reached at step 3 or 4 of your worked answer, is correct. The equations would be inconsistent if the rank of the augmented matrix was greater than the rank of the coefficient matrix. These ranks become obvious once you have put the equations in row echelon form (which you have done at step 3). At that stage it is clear that the rank of the coefficient matrix is 3 (= the rank of the augmented matrix) regardless of the values of [MATH]a[/MATH] and [MATH]b[/MATH].

 

[Wen Qualto]

Your conclusion, which you could have reached at step 3 or 4 of your worked answer, is correct. The equations would be inconsistent if the rank of the augmented matrix was greater than the rank of the coefficient matrix. These ranks become obvious once you have put the equations in row echelon form (which you have done at step 3). At that stage it is clear that the rank of the coefficient matrix is 3 (= the rank of the augmented matrix) regardless of the values of [MATH]a[/MATH] and [MATH]b[/MATH].

Thank you very much! That was very helpful!

 

[ETK]

Perhaps there is a typo in the question?
If instead, the second equation was:
[MATH]x_1+2x_2-x_3+x_4=1[/MATH],
then the equations would be inconsistent if [MATH]b=-3[/MATH] and [MATH]a \neq 4[/MATH].
Do you agree?

 

[Wen Qualto]

Perhaps there is a typo in the question?
If instead, the second equation was:
[MATH]x_1+2x_2-x_3+x_4=1[/MATH],
then the equations would be inconsistent if [MATH]b=-3[/MATH] and [MATH]a \neq 4[/MATH].
Do you agree?

I double checked, I wrote the correct system, there could be a typo in the exercise I guess. The reason I was a bit sceptical was that there is a b) and c) part to the question that becomes a bit odd. They are solvable but the answers become a bit strange.