- 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**If

**AB**exists, it does not mean**BA**exists

To find the product of two matrices, multiply the elements in each row of the first matrix by the elements in each column of the second matrix

## Determinants

The **determinant** of a square matrix is a scalar value that represents the matrix. It is only possible for square matrices.

The determinant of a matrix,

**M**, is written as**det M**, or as**|M|**A

**singular matrix**has a determinant of zero.A

**non-singular matrix**has a determinant that is not zero.

To find the determinant of a 2x2 matrix:

To find the determinant of a 3x3 matrix:

## Inverse Matrices

The inverse of a matrix **M** is the matrix **M¯¹**

M M¯¹ = M¯¹ M = 1

### Finding the Inverse of a 2x2 Matrix

Find the determinant

Write out the matrix where

*a*and*d*are swapped, and both*b*and*c*are multiplied by -1Multiply by one over the determinant

If A and B are non-singular matrices, (AB)¯¹ = B¯¹A¯¹

### Finding the Inverse of a 3x3 Matrix

Find the determinant

Replace each element with it matrix of minors & find the determinants

Form the matrix of co-factors (switch every other sign)

Find the transpose of the matrix of co-factors by switching rows and columns

Multiply this by one over the determinant

Since this is such a complicated thing to do, unless specifically needed, use the calculator function.

On the __CASIO ClassWiz fx-991EX__:

Click

**MODE**Click

**4**: MatrixClick

**1**: Define Matrix AClick

**3 twice**to set size of matrixFill in Matrix

Click

**AC**Click

**OPTN**CLICK

**3**: Matrix AClick

**x****¯¹**Click =

## Matrix Systems & Geometry

It is possible to use matrices and their inverses to easily solve simultaneous equations.

This example is **consistent**, meaning there is at least one set of values that satisfies all three equations simultaneously. The determinant is not zero, making the matrix non-singular.

This is especially useful for 3D geometry, and the equations of planes:

## Linear Transformations

A linear transformations are transformations with specific properties:

they are made up of linear x, y and z terms only

they have no non-variable terms

they map the origin onto itself

they can be represented by matrices

Points and lines that do not change in a transformation are called invariant points/lines

The origin is always an invariant point.

Every point on an invariant line is an invariant point.

### Reflections

The line of reflection is always an invariant line.

In two-dimensions, there are standard reflections for the coordinate axes and y=±x.

In three-dimensions, reflections ca happen in planes:

For a linear transformation with matrix M, if the determinant is negative, the shape has been reflected

### Rotations

For rotations about the origin, the only invariant point is the origin.

In 3D, rotations can be about an entire axis:

### Enlargements & Stretches

If a = b, the transformation is an enlargement of scale factor a.

For a stretch that is only in the x-direction, the y-axis is an invariant line.

For a stretch that is only in the y-direction, the x-axis is an invariant line.

For stretches in both directions, there are no invariant lines and the only invariant point is the origin.

For a linear transformation with matrix M, the determinant is the area scale factor.

### Successive & Inverse Transformations

It is possible to apply multiple transformations, one after the other. In this instance,** order of matrix multiplication is particularly important.**

For two linear transformations, represented by the matrices **P** and **Q **respectivley:

PQ represents the transformation Q followed by P

A linear transformation with matrix **A** can be undone with the transformation represented by the inverse matrix, **A¯¹**.