# Matrices

In mathematics, a **matrix** (plural **matrices**) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimension of the matrix below is `2 × 3` (read "two by three"), because there are two rows and three columns:

```
| 1  9 -13 |
| 20 5 -6  |
```

![An `m × n` matrix](https://upload.wikimedia.org/wikipedia/commons/b/bf/Matris.png)

An `m × n` matrix: the `m` rows are horizontal, and the `n` columns are vertical. Each element of a matrix is often denoted by a variable with two subscripts. For example, <i>a<sub>2,1</sub></i> represents the element at the second row and first column of the matrix

## Operations on matrices

### Addition

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

![Matrices addition](https://www.mathsisfun.com/algebra/images/matrix-addition.gif)

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

### Subtracting

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

![Matrices subtraction](https://www.mathsisfun.com/algebra/images/matrix-subtraction.gif)

### Multiply by a Constant

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

![Matrices multiplication be a constant](https://www.mathsisfun.com/algebra/images/matrix-multiply-constant.gif)

### Multiplying by Another Matrix

To multiply a matrix by another matrix we need to do the [dot product](https://www.mathsisfun.com/algebra/vectors-dot-product.html) of rows and columns.

To work out the answer for the **1st row** and **1st column**:

![Matrices multiplication - 1st step](https://www.mathsisfun.com/algebra/images/matrix-multiply-a.svg)

Here it is for the 1st row and 2nd column:

![Matrices multiplication - 2st step](https://www.mathsisfun.com/algebra/images/matrix-multiply-b.svg)

If we'll do the same for the rest of the rows and columns we'll get the following resulting matrix:

![Matrices multiplication - Result](https://www.mathsisfun.com/algebra/images/matrix-multiply-c.svg)

### Transposing

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

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

![Transposing](https://www.mathsisfun.com/algebra/images/matrix-transpose.gif)

## References

- [Matrices on MathIsFun](https://www.mathsisfun.com/algebra/matrix-introduction.html)
- [Matrix on Wikipedia](https://en.wikipedia.org/wiki/Matrix_(mathematics))