How to Tell if a System is Inconsistent

Consider the following 4 by 5 matrix, F, which represents the augmented matrix for a system of 4 equations in 4 variables:

> F:=matrix(4,5,[4,2,-5,-5,-3,4,-3,-2,-5,2,-5,4,0,4,5,-8,6,4,10,0]);

Let's obtain a Row-Echelon form:

> gausselim(F);

This is inconsistent, because it is saying "0=4" which isn't true. Therefore the system of equations has no solution.

Now consider another 4 by 5 matrix F2, which again represents the augmented matrix for a system of 4 equations in 4 variables:

> F2:=matrix(4,5,[4,2,-5,-5,-3,4,-3,-2,-5,2,-5,4,0,4,5,-8,6,4,0,-4]);

Let's obtain a Row-Echelon form:

> gausselim(F2);

This system is consistent. Don't get confused! It is not saying "-10=0", rather, it is saying -10*x_4 = 0, or in other words, x_4 = 0, which is perfectly okay. The system can be solved completely, using the Gauss-Jordan Method:

> gaussjord(F2);

>