Here are some example matrices and how to interpret them.  Assume that we are solving a system of equations involving 3 variables (x, y, and z) using our calculator's reduced row echelon feature (rref()).  Here are 3 types of results we could get:

Matrix 1:		In this matrix the system of equations is independent which means that there is one set of solutions.  x=2, y=3, and z=4.
Matrix 2:		In this matrix the system of equations is inconsistant meaning that there are no solutions.  You can tell this by reading the last line.  It says 0z=4 meaning 0=4 which of course can't be true.
Matrix 3:		In this matrix the system of equations is dependent, meaning 2 of the answers depends on the value picked for the third answer.  I.e. there are infinite solutions.  You can tell this by reading the last line which says 0z=0 which of course is true no matter what value you put in for z. Thus we have x+z=2 which leads to x=2-z,  y+2z=3 which leads to y=3-2z, and z= any real.

Hope all this helps.  Let me know if you have questions.