# Matrices

A Matrix is an array of numbers:

(This one has 2 Rows and 3 Columns)

We talk about one **matrix**, or several **matrices**.

There are many things we can do with them ...

## Adding

To add two matrices: add the numbers in the matching positions:

3+4=7 | 8+0=8 |

4+1=5 | 6−9=−3 |

The two matrices must be the same size, i.e. the rows must match in size, and the columns must match in size.

Example: a matrix with **3 rows** and **5 columns** can be added to another matrix of **3 rows** and **5 columns**.

But it could not be added to a matrix with **3 rows** and **4 columns** (the columns don't match in size)

## Negative

The negative of a matrix is also simple:

−(2)=−2 | −(−4)=+4 |

−(7)=−7 | −(10)=−10 |

## Subtracting

To subtract two matrices: subtract the numbers in the matching positions:

3−4=−1 | 8−0=8 |

4−1=3 | 6−(−9)=15 |

*Note: subtracting is actually defined as the addition of a negative matrix: A + (−B)*

## Multiply by a Constant

We can multiply a matrix by a **constant** *(the value 2 in this case)*:

2×4=8 | 2×0=0 |

2×1=2 | 2×−9=−18 |

We call the constant a **scalar**, so officially this is called "scalar multiplication".

## Multiplying by Another Matrix

To **multiply two matrices together** is a bit more difficult ... read Multiplying Matrices to learn how.

## Dividing

And what about division? Well we **don't** actually divide matrices, we do it this way:

A/B = A × (1/B) = A × B^{-1}

where **B ^{-1}** means the "inverse" of B.

So we don't divide, instead we **multiply by an inverse**.

And there are special ways to find the Inverse, learn more at Inverse of a Matrix.

## Transposing

To "transpose" a matrix, swap the rows and columns.

We put a "T" in the top right-hand corner to mean transpose:

## Notation

A matrix is usually shown by a **capital letter** (such as A, or B)

Each entry (or "element") is shown by a **lower case letter** with a "subscript" of **row,column**:

_{1,1}a

_{1,2}a

_{1,3}a

_{2,1}a

_{2,2}a

_{2,3}

## Rows and ColumnsSo which is the row and which is the column? - Rows go
**left-right** - Columns go
**up-down**
To remember that rows come before columns use the word a |

### Example:

Here are some sample entries:

b_{1,1} = 6 *(the entry at row 1, column 1 is 6)*

b_{1,3} = 24 *(the entry at row 1, column 3 is 24)*

b_{2,3} = 8 *(the entry at row 2, column 3 is 8)*