Solving system of linear equations

system of linear equations ->       Ax=b  (m by n coefficient matrix A)

-> [A | b]  (the augmented matrix)

---------------------------------------->         (three  possible  outcomes)
row elementary operations ,     column permutation,   dropping redundant rows

 
0|  0     --->   0=0  (every x is a solution)

0 | 1  --->     0 =1   (there are no   solutions)

yT      zT    =
 --------       ---->   y=-Cz+d   (variables in z, if any, take arbitrary values; k <= m)
1k      C   | d