A consequence of our discussion on the structure of solutions is the criterion for the uniqueness of the solution. Combining the following facts

- uniqueness ⇔ no freedom (i.e., no free variables)
- free variable ⇔ nonpivot column
- nonfree variable ⇔ pivot column

we have

A consistent system of linear equations has a unique solution

⇔ All variables are nonfree

⇔ All columns of the coefficient matrix are pivot

Example In a previous example, we saw that the system

x_{1} |
+ 3x_{2} |
+ 2x_{3} |
+ x_{5} |
= | 0 | |

- x_{1} |
- x_{2} |
- x_{3} |
+ x_{4} |
= | 1 | |

4x_{2} |
+ 2x_{3} |
+ 4x_{4} |
+ 3x_{5} |
= | 2 | |

x_{1} |
+ 3x_{2} |
+ 2x_{3} |
- 2x_{4} |
= | 0 |

is consistent. Moreover, in the row echelon form of the coefficient matrix,

[ | 1 | 3 | 2 | 0 | 1 | ] |

0 | 2 | 1 | -1 | 0 | ||

0 | 0 | 0 | 2 | 1 | ||

0 | 0 | 0 | 0 | 0 |

[col 3] and [col 5] are not pivot. Therefore the solution is not unique.

The following system is obtained by taking away `x`_{3}
and `x`_{5}.

x_{1} |
+ 3x_{2} |
= | 0 | |

- x_{1} |
- x_{2} |
+ x_{4} |
= | 1 |

4x_{2} |
+ 4x_{4} |
= | 2 | |

x_{1} |
+ 3x_{2} |
- 2x_{4} |
= | 0 |

The system was also studied before and was known to be consistent. In the row echelon form of the coefficient matrix,

[ | 1 | 3 | 0 | ] |

0 | 2 | -1 | ||

0 | 0 | 2 | ||

0 | 0 | 0 |

all columns are pivot. Therefore the solution is unique.

We remark that the criterion for the uniqueness does not involve the right side.
In other words, if both ** Ax** =

The uniqueness is independent of the right side. Moreover,

A consistent system ** Ax** =

⇔ The homogeneous system

Example The following system

x_{1} |
+ 3x_{2} |
+ 2x_{3} |
+ x_{5} |
= | 0 | |

- x_{1} |
- x_{2} |
- x_{3} |
+ x_{4} |
= | 1 | |

4x_{2} |
+ 2x_{3} |
+ 4x_{4} |
+ 3x_{5} |
= | 4 | |

x_{1} |
+ 3x_{2} |
+ 2x_{3} |
- 2x_{4} |
= | -2 |

has the obvious solution `x`_{1} = `x`_{2} =
`x`_{3} = `x`_{5} = 0, `x`_{4} = 1.
In the previous example, we saw that another system with the same coefficient matrix has non-unique solutions.
Therefore this system also has non-unique solutions. In other worlds,
the system must have solutions other than the obvious one. Similarly, the homogeneous system

x_{1} |
+ 3x_{2} |
+ 2x_{3} |
+ x_{5} |
= | 0 | |

- x_{1} |
- x_{2} |
- x_{3} |
+ x_{4} |
= | 0 | |

4x_{2} |
+ 2x_{3} |
+ 4x_{4} |
+ 3x_{5} |
= | 0 | |

x_{1} |
+ 3x_{2} |
+ 2x_{3} |
- 2x_{4} |
= | 0 |

must have nontrivial (i.e., nonzero) solutions. On the other hand, the system

x_{1} |
+ 3x_{2} |
+ 2x_{3} |
+ x_{5} |
= | 23 | |

- x_{1} |
- x_{2} |
- x_{3} |
+ x_{4} |
= | -45 | |

4x_{2} |
+ 2x_{3} |
+ 4x_{4} |
+ 3x_{5} |
= | 78 | |

x_{1} |
+ 3x_{2} |
+ 2x_{3} |
- 2x_{4} |
= | 12 |

does not have an obvious solution. Since it has the same coefficient matrix as before, we conclude there are two possibilities.

- The system has no solution
- The system has more than one solutions

Of course, to find out which case exactly happens, one has to do more computations (in order to determine the consistency).

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