Help solving a linear system question #2

[dumbledore]

Hi,

I need to answer the question as to what the pivot columns are for the following matrix representing a linear system of equations as well as whether the system is consistent or inconsistent.

The augmented matrix for the system is:

-2 0 3 1 | -1
0 -1 1 1 | 2
2 0 -1 -1 | 1
-2 -1 4 2 | 0

I performed row reduction using the following operations:

R1/-2
R3+R4-->R3
2R1+R4-->R4
R2/-1
R2+R3-->R3
R2+R4-->R4
R3/2
R2+R3-->R2
3/2R3+R1-->R1

I end with the following augmented matrix:

1 0 0 -11/4 | -3/4
0 1 0 -5/2 | -5/2
0 0 1 -3/2 | -1/2
0 0 0 0 | -1

Based on this I conclude that the system is inconsistent because of the last row we have 0=-1. I also conclude that the pivot columns are columns 1,2, and 3.



Can someone please check to see if I conducted the problem correctly and ended up with the correct matrix?

Thank you

 

[HallsofIvy]

I did this a little differently (not trying to get "1" in the pivot positions) and arrived at
\(\displaystyle \begin{bmatrix}-2 & 0 & 3 & 1 & -2 \\ 0 & -1 & 1 & 1 & 2 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1\end{bmatrix}\)
but that gives the same result: The pivots are on the diagonal for the first three rows and the system is inconsistent, just as you say.

 

[dumbledore]

I did this a little differently (not trying to get "1" in the pivot positions) and arrived at
\(\displaystyle \begin{bmatrix}-2 & 0 & 3 & 1 & -2 \\ 0 & -1 & 1 & 1 & 2 \\ 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1\end{bmatrix}\)
but that gives the same result: The pivots are on the diagonal for the first three rows and the system is inconsistent, just as you say.

Thanks a lot! I appreciate it