- A-Level Further Maths

# Matrices & Linear Transformations

A matrix is a system of elements within a pair of brackets. The size of the matrix is given as the number of rows and columns in it.

A

**square matrix**is one where there is an equal number of rows and columnsA

**zero matrix**is one in which all the elements are zeroAn

**identity matrix**is a square matrix where the all the values on the diagonal from top left to bottom right are 1, and all other values are zero. This is noted as capital i,**I**with a subscript number afterwards to show its size.

## Working with Matrices

### Adding and Subtracting

To add or subtract matrices, you add or subtract the corresponding elements in each matrix.

You can only add or subtract matrices of the same size

### Multiplying by a Scalar

To multiple a vector by a scalar, simply multiple each element in the vector by the scalar. This can be factorised out, too.

### Multiplying One Vector by Another

Matrix multiplication is only possible if the **number of columns in the first matrix equals the number of rows in the second matrix**. The result of the multiplication is known as the **product matrix**, and will have the same number of rows as the first matrix and the same number of columns as the second matrix.

Order of matrix multiplication matters

For two matrices, **A** and **B**:

In general,

<**AB ≠ BA**