In systems of linear equations, the notations for the variables are not essential.

Example The essential information of the following system

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

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

2x_{1} |
+ 2x_{2} |
+ x_{3} |
= | 6 |

is contained in the matrix

[ | 3 | 1 | -1 | 2 | ] |

1 | -1 | 1 | 2 | ||

2 | 2 | 1 | 6 |

made up of the coefficients on the left side and the numbers on the right side. Conversely, the system can be recovered from the matrix. For example, the following matrix

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

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

-2 | 0 | -5 | 1 |

comes from the following system (provided `x`_{1},
`x`_{2}, `x`_{3} are chosen as the variables).

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

x_{2} |
- x_{3} |
= | -2 | |

- 2x_{1} |
- 5x_{3} |
= | 1 |

In general, the essential information of a system of linear equations

a_{11}x_{1} |
+ a_{12}x_{2} |
+ ... + a_{1n}x_{n} |
= | b_{1} |

a_{21}x_{1} |
+ a_{22}x_{2} |
+ ... + a_{2n}x_{n} |
= | b_{2} |

... | ... | ... | ||

a_{m1}x_{1} |
+ a_{m2}x_{2} |
+ ... + a_{mn}x_{n} |
= | b_{m} |

is contained in the augmented matrix

[] = [A b |
a_{11} |
a_{12} |
... | a_{1n} |
b_{1} |
] |

a_{21} |
a_{22} |
... | a_{2n} |
b_{2} |
||

: | : | : | : | |||

a_{m1} |
a_{m2} |
... | a_{mn} |
b_{m} |

which is made up of the coefficient matrix

= [A |
a_{11} |
a_{12} |
... | a_{1n} |
] |

a_{21} |
a_{22} |
... | a_{2n} |
||

: | : | : | |||

a_{m1} |
a_{m2} |
... | a_{mn} |

and the right side vector.

= [b |
b_{1} |
] |

b_{2} |
||

: | ||

b_{m} |

We also denote the left side of the system by

= [Ax |
a_{11}x_{1}
+ a_{12}x_{2}
+ ... + a_{1n}x_{n} |
] |

a_{21}x_{1}
+ a_{22}x_{2}
+ ... + a_{2n}x_{n} |
||

: | ||

a_{m1}x_{1}
+ a_{m2}x_{2}
+ ... + a_{mn}x_{n} |

so that the system of linear equations is ** Ax** =

Example Consider the system

x |
+ 2y |
= | 3 |

4x |
+ 5y |
= | 6 |

7x |
+ 8y |
= | 9 |

10x |
+ 11y |
= | 12 |

The coefficient matrix is

= [A |
1 | 2 | ] |

4 | 5 | ||

7 | 8 | ||

10 | 11 |

the right side vector is

= [b |
3 | ] |

6 | ||

9 | ||

12 |

And the augmented matrix is

[] = [A b |
1 | 2 | 3 | ] |

4 | 5 | 6 | ||

7 | 8 | 9 | ||

10 | 11 | 12 |

Finally, we note the following correspondences between the augmented matrix and the system.

columns of ` A` ⇔ variables of the system

rows of [

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