Are my calculations correct?

Question

Solve the following system using Gaussian elimination:

$ x + 4y + z = 0\\ 4x + 13y + 7z = 0\\ 7x + 22y + 13z = 1$

This is what I have done:

Augmented Matrix:    1 4   1    0
                   4  13   7    0  
                   7   22   13   1  

$ R_2 - 4R_1 = R_2$ and $R_3 - 7R_1 = R_3$

                   1    4    1    0
                   0   -3    3    0
                   0   -6    6    1

$-1/3R_2 = R_2$

                   1    4    1    0
                   0    1   -1    0
                   0   -6    6    1

$R_3 + 6R_2 = R_3$

                   1    4    1    0
                   0    1   -1    0
                   0    0    0    1

By what I understand, the last row means $0x + 0y + 0z = 1$ This is incorrect so this means that the system is inconsistent Have I made a mistake somewhere with my calculations or am I misunderstanding something? Or is the system really inconsistent and does that mean that it has no valid solution?

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Answer 1

Yes, the system is indeed inconsistent. The rank of the original matrix (non-augmented) is $2$, which means that the system either has infinitely many solutions or it has zero solutions. Your manipulation shows that it has zero solutions.