Existence and Uniqueness

Consider the augmented matrix:

The leading one’s are in first, third, and fifth columns.

• The pivot columns (leading ones) of A are the first, third, and fifth columns
• The corresponding variables of the system $A\mathbf{x}=\mathbf{b}$. are $x_1,x_2,\text{and}{x}_5$ Variables that correspond to a pivot are basic variables.
• Variables that are not basic are free variables. They can take any value.
• The free variables are $x_2$ and $x_4$. Any choice of the free variables leads to a solution of the system.

Unique solution exists if and only if there are no free variables.

A linear system is consistent if it has at least one solution.

Note

If you see something like this (b is a non-zero number):

$$\left[\begin{array}{cccccc}0& 0& 0& ...& 0& b\end{array}\right]$$

RUN you don’t have a Consistent System.