Aware of my minimal knowledge of linear algebra, I was seeing the power of the Gauss-Jordan elimination method. In particular, it can be used for:
solve the linear systems $A\,X = b$;
calculate the rank of a matrix $A$ $m \times n$;
calculate the determinant of a matrix $A$ $n \times n$;
calculate the inverse of a matrix $A$ $n \times n$ with $\det(A)\ne 0$, considering a matrix $[A \; I]$ which becomes $[I \; A^{-1}]$;
to extract a vector base of the subspace generated by the column vectors of a matrix $A$ $m \times n$;
to extract a vector base of the subspace generated by the kernel vectors of a matrix $A$ $m \times n$, considering a matrix $[A^t\,I]$ which becomes $[B \; C]$ (excellent method to extract, therefore, also the eigenvectors).
Are there other applications of this method in the field of linear algebra besides those listed?