A linear algebra problem! Why isn't C correct? Every column has a pivot (the condition for one-one) in C

[imvishcv]

 

[blamocur]

My guess is that C is not good enough because Gaussian elimination is incomplete: the 3rd row can be zeroed by subtracting scaled 4th row. On the other hand it seems to satisfy the definition of column echelon form in Wikipedia (https://en.wikipedia.org/wiki/Row_echelon_form).