I have been trying to find the inverse of a matrix using Gauss method and I want to know suppose what happens if I don't get the "1" in reduced matrix on the left? Does it mean that the inverse doesn't exist?
$$ \begin{bmatrix} 1 & 0 & 0 : 1 & 2 & 1 \\ 0 & 1 & 0 : 0 &-3 & 0 \\ 0 & 0 & 0 : 1 & 0 & 1 \\ \end{bmatrix} $$