**Introduction to Matrices**

What is a **matrix**?

What is a **square matrix**?

How do we do arithmetic with matrices? (which ones can we do it with?)

What is an **identity matrix**?

What is a **zero matrix**?

**Worked Examples**

1.) For the matrices **A** = , **B **= , **C **= , **D** = , **E** = and **F** = , find, where possible:

**A**–**E****C**+**D****E**+**A**–**B****F**+**D****D**–**C**- 4
**F**

2.) Calculate

3.) Calculate

**Exercise**

**Answers**

**Introduction to Row Operations and Inverse Matrices**

What are the three** row operations**?How can we find the **inverse** of a square matrix of any size? ([A|I] = [I|A^{-1}] (we will find out why in FP2 when we talk about augmented matrices and think of matrices as a way of solving systems of equations).

What is **row echelon form**? What is **reduced row echelon form**?

What are **singular** and **non-singular **matrices?

I will find the inverse of:

You can find the inverse of:

With a 2×2 matrix we can use a formula as a quick alternative to find the inverse:

If **M** = , then **M ^{-1}** =

N.B. (AB)^{-1} = B^{-1}A^{-1}

**Worked Examples**

1.) If **B** = , then find **B ^{-1}**.

2.) If **M **= and **N **= , then calculate:

**M**^{-1}**N**^{-1}**MN****NM**- (
**MN**)^{-1} - (
**NM**)^{-1} **M**^{-1}**N**^{-1}**N**^{-1}**M**^{-1}

For a 3×3 matrix we can also use a slightly more complicated formula to find the inverse using **cofactors**. We can come back and look at this and compare it with the Gaussian method after we have learned about determinants (further down).

**Worked Example**

Find the inverse of the matrix **N** without using a calculator, where **N** =

**Exercise**

**Answers**

**Introduction to Determinants**

**ad-bc **is the determinant of:

Each of the terms is called a **minor**

**Worked Examples**

1.) Find the determinant of N =

**Determinants of larger matrices**

If we reduce a matrix to **row echelon form**, then the determinant is the product of the values in the leading diagonal.

The row operations **switching rows** and **adding rows** do not affect the determinant.

The row operation **multiplying a row by a scalar** increases the determinant by that scalar multiple (so if we do this when reducing a matrix, we must divide the determinant calculated by this number.

**Det (A) = 0** means that the matrix is singular.

Transposing a matrix also doesn’t affect the determinant.

**Worked Example**

Given that **P** = and **Q** = , find:

- det
**P**; - det
**Q**; - det
**PQ**. What do you notice?

**Alternative method for finding determinants**

Find the determinant of the matrix

**Exercise**

**Answers**

**Introduction to Matrix Transformations**

Consider (pre-)multiplying the position vector by the following **transformation matrices:** , , .

Of course a transformation affects not just a single point, but the whole plane, e.g. consider effect on rectangle: A:(1,1), B:(1,2), C;(3,1), D:(3,2)

Any point in the plane unchanged by a transformation is an **invariant point**. The **origin** is always an invariant point.

Any line unchanged by a transformation is an **invariant line **(N.B. points on this line may move to another point, but only within the line). For an **enlargement matrix**, the x-axis and y-axis are both invariant lines.

When enlarging a shape, the area of the shape is increased by the determinant of the transformation matrix.

What transformation will the following two matrices effect?: , .

If you imagine reflecting in the y-axis and then reflecting in the x-axis, the combined transformation is effectively a rotation by 180 degrees. So the transformation matrix for this is: .

We want a rotation matrix that will work for any different angle.

Consider the point (x,0)

In order that changing the angle (theta) will rotate anticlockwise by the specified amount, we want:

So ax = xcos𝜽 and cx = xsin𝜽

So a = cos𝜽 and c = sin𝜽

As we don’t want an enlargement |A| = ad-bc = 1 and so dcos𝜽 – bsin𝜽 = 1.

From Pythagoras’ Theorem, we know that one solution for this is b = -sin𝜽, d = cos𝜽.

So our rotation matrix is:

For instance, a 90º clockwise rotation would be achieved with

As well as **enlargement**, **reflection**, and **rotation**, we also have **shearing**.

In a shear, all points in the plane are stretched in the same direction, but the extent of the stretch is proportional to the distance from the axes. Below shows the effect of a transformation of on the unit square:

Note 1: The inverse of a transformation matrix always undoes the transformation

Note 2: Any transformations can be combined by multiplying the matrices together

**Invariant Lines**

Invariant lines must pass through the origin, so they have the form y=mx

We can find m by solving the following, which checks that if the object is on the invariant line, the the image is also on this line:

**Worked Examples**

**Exercise & Combined Exercises**

**Answers**

**Past Exam Paper Questions**

Summer 2020 Paper 13:

Summer 2020 Paper 13 Mark Scheme

Winter 2020 Paper 11:

Winter 2020 Paper 11 Mark Scheme

Winter 2020 Paper 12