Let `M`(`m`, `n`) be the collection of all `m` by `n` matrices.
The zero matrix is

=[O |
0 | 0 | ... | 0 | ] |

0 | 0 | ... | 0 | ||

: | : | : | |||

0 | 0 | ... | 0 |

Like vectors, matrices of the same size may be added.

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

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

: | : | : | : | : | : | : | : | : | |||||||

a_{m1} |
a_{m2} |
... | a_{mn} |
b_{m1} |
b_{m2} |
... | b_{mn} |
a_{m1}+b_{m1} |
a_{m2}+b_{m2} |
... | a+_{mn}b_{mn} |

A number (scalar) can also be multiplied to matrices

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

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

: | : | : | : | : | : | |||||

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

Moreover, we may combine the two operations and form the linear combinations of matrices (of the same size).

Example Let

= [A |
1 | 1 | -2 | ], = [B |
-1 | 0 | 1 | ], = [C |
0 | 1 | 0 | ], = [D |
0 | 1 | ] |

3 | 0 | 1 | -2 | 1 | 1 | 1 | 0 | 0 | 1 | 2 |

Then

+ A = [B |
1 + (-1) | 1 + 0 | (-2) + 1 | ] = [ | 0 | 1 | -1 | ] |

3 + (-2) | 0 + 1 | 1 + 1 | 1 | 1 | 2 |

+ A + B = [C |
1 + (-1) + 0 | 1 + 0 + 1 | (-2) + 1 + 0 | ] = [ | 0 | 2 | -1 | ] |

3 + (-2) + 1 | 0 + 1 + 0 | 1 + 1 + 0 | 2 | 1 | 2 |

2 = [A |
2×1 | 2×1 | 2×(-2) | ] = [ | 2 | 2 | -4 | ] |

2×3 | 2×0 | 2×1 | 6 | 0 | 2 |

-3 = [D |
(-3)×0 | (-3)×1 | ] = [ | 0 | -3 | ] |

(-3)×1 | (-3)×2 | -3 | -6 |

3 - 4A + 2B = [C |
3 + 4 + 0 | 3 + 0 + 2 | - 6 - 4 + 0 | ] = [ | 7 | 5 | -10 | ] |

9 + 8 + 2 | 0 - 4 + 0 | 3 - 4 + 0 | 19 | -4 | -1 |

On the other hand, expressions such as ` A` +

*Euclidean vectors can be considered as matrices of special sizes*.
Specifically, a vertical `n`-dimensional euclidean vector is an `n` by 1 matrix,
and the euclidean space **R**^{n} is identified with `M`(`n`, 1).
On the other hand, if we present vectors horizontally, then **R**^{n} is identified with `M`(1, `n`).

Although euclidean vectors may be presented either vertically or horizontally,
you are strongly advised to use vertical vectors when they are mixed with matrices.
For example, in expressing a system of linear equation as ** Ax** =

*Conversely, matrices can be considered as an (ordered) collection of vectors*.
Specifically, an `m` by `n` matrix is equivalent to `n` euclidean column vectors of dimension `m`.

` A` = [

Similarly, we may also consider row vectors, so that an `m` by `n` matrix is equivalent to
`m` euclidean row vectors of dimension `n`.

Example The matrix

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

4 | 5 | 6 |

can be considered as three 2-dimensional column vectors

=
[Aa_{1} a_{2} a_{3}],
a_{1} = [ |
1 | ], a_{2} = [ |
2 | ], a_{3} = [ |
3 | ] |

4 | 5 | 6 |

It can also be considered as two 3-dimensional row vectors.

=
[A |
b_{1} |
], b_{1} = (1, 2, 3), b_{2} = (4, 5, 6) |

b_{2} |

The relations between the vectors and the matrices are compatible with addition
and scalar multiplication. For example, for three column matrices
` A` = [

` A` +

Because of this compatibility, the properties for the addition and scalar multiplication of vectors are also hold for matrices.