Further Maths

• There are roughly 600,000 GCSE students in the country at any one time

• Only 26,000 students sat AQA Further Maths last year

• That means by just sitting the exam you are potentially in the most able 4% of the country

• 93% of that 4% get a C or above

Further Maths

• What does that mean for you?

• There will be things in these sessions you may not have been taught.

• They may be things that you are already expert in.

Further Maths

• You will be given a set of past papers and a booklet of questions.

• If you are already and expert on a topic you can work on these questions (without disturbing the session)

• Please bring the past papers to each session (especially important on Friday)

Further Maths

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• All of the resources and powerpoints will be available through the blog.

StarterThese tables show information on items sold in 2 different shops over

several days. Summarise the information into a single table.

Mathematically, this is the start of ‘Matrix Algebra’

It is a method computers use to add up large amounts of data

It is also used in computer animation, as matrices can

transform the shapes of objects!

Shop A TVs Radios PhonesDAY 1 7 3 12DAY 2 6 2 8DAY 3 7 2 9DAY 4 10 4 11

Shop B TVs Radios PhonesDAY 1 8 4 14DAY 2 3 6 10DAY 3 9 5 11DAY 4 12 5 12

[ 7 3 126 2 87 2 910 4 11 ]+[ 8 4 14

3 6 109 5 1112 5 12]¿ [15 7 26

9 8 1816 7 2022 9 23 ]

We can use matrices to represent the information above…

Matrix Algebra (+,-)

Matrix AlgebraTo begin with, you need to

know how to solve problems involving the addition and

subtraction of matrices, and be able to state the ‘order’ of

a matrix (its dimensions)

The order of a matrix is (n x m) where n is the number of rows and m is the number of

columns

Write the dimensions of the following matrices

[2 −11 3 ] b) [1 0 2 ]

d)[ 4−1] [ 3 2−1 10 −3]

2 rows 2 columns The matrix is 2 x 2

1 row 3 columns The matrix is 1 x 3

2 rows 1 column The matrix is 2 x 1

3 rows 2 columns The matrix is 3 x 2

Matrix AlgebraTo begin with, you need to

know how to solve problems involving the addition and

subtraction of matrices, and be able to state the ‘order’ of

a matrix (its dimensions)

You can add and subtract matrices only when they have

the same dimensions

𝑨=[ 5 7 4−6 −2 3 ]

𝑩=[ 8 −2 0−3 8 −1]

Calculate A + B

[ 5 7 4−6 −2 3 ]+[ 8 −2 0

−3 8 −1]

¿ [¿ ]26−94513

Calculate A - B

[ 5 7 4−6 −2 3 ]−[ 8 −2 0

−3 8 −1 ]

¿ [¿ ]4−10−349−3

Matrix Algebra (multiplication 1)

Matrix Algebra (2)You need to be able to multiply a matrix by a

number, as well as another matrix

Calculate:a) 2A

b) -3A

𝑨=[ 5 2−4 0]

𝑨=[ 5 2−4 0]a)

2 𝑨=[ 10 4−8 0 ]

𝑨=[ 5 2−4 0]b)

−3 𝑨=[−15 −612 0 ]

Just multiply each part by

2

Just multiply each part by -3

So to multiply a matrix by a number, you just multiply each part in the matrix separately

Matrix Algebra (2)You need to be able to multiply a matrix by a

number, as well as another matrix

To multiply matrices together, multiply each

ROW in the first, by each COLUMN in the second (like

in the starter)

Remember for each row and column pair, you need

to sum the answers!

a) Calculate the following

[2 5 3 ][461 ] Multiply each number in the row with the

corresponding number in the column

(2×4 )+(5×6 )+(3×1 )¿ 41

The values of x and y in these pairs of Matrices are the same. Calculate what x and y must be!

[ 𝑥 𝑦 ] [53]=[20 ]

[ 𝑦 −2 ] [ 2𝑥 ]=[−24 ]

5 𝑥+3 𝑦=20

2 𝑦−2 𝑥=−24

10 𝑥+6 𝑦=40

10 𝑦−10 𝑥=−120

16 𝑦=−80𝑦=−5𝑥=7

As an equation

As an equation Multiply by

2

Multiply by 5

Add the two equations together

Divide by 16

Then find x

[ 𝑥 𝑦 ] [53]=[20 ] [ 𝑦 −2 ] [2𝑥 ]=[−24 ]

Matrix Algebra (multiplication 2)

Matrix Algebra (3)Multiplying Matrices together

When you have more difficult matrices, follow these steps:

Write the order of the matrices, and hence the order of the answer.

Take the first row of the first matrix, and multiply it by the first column of the second (as you have been doing up until now). Remember these will be added together (write the sum out first…)

Then continue using the first row and multiply by the next column (if there is one). Write the answer down next to the first (horizontally)

Once you have used the first row with all the columns, repeat the process but with the second row, (if there is one!) writing these answers below the first set

Continue until you have used all the rows with all the columns

Then calculate each sum – it will already be set out in the correct position!

Lets see an example!

Calculate the following:

[5 6 ][34 12]

1 x 2 2 x 2 1 x 2

¿ [¿ ]

(5×3 )+(6×4)(5×1 )+(6×2)

¿ [3 9 17 ]

Matrix Algebra (3)Multiplying Matrices together

When you have more difficult matrices, follow these steps:

Write the order of the matrices, and hence the order of the answer.

Take the first row of the first matrix, and multiply it by the first column of the second (as you have been doing up until now). Remember these will be added together (write the sum out first…)

Then continue using the first row and multiply by the next column (if there is one). Write the answer down next to the first (horizontally)

Once you have used the first row with all the columns, repeat the process but with the second row, (if there is one!) writing these answers below the first set

Continue until you have used all the rows with all the columns

Then calculate each sum – it will already be set out in the correct position!

Lets see an example!

Calculate the following:

[ 1 3−5 0 ][3 7

4 −1]2 x 2 2 x 2 2 x 2

¿ [¿ ]

(1×3 )+(3×4 )

¿ [ 15 4−15 −35 ]

(1×7 )+(3×−1)(−5×3 )+(0×4)(−5×7 )+(0×−1)

Practise

Answers

Practise

Extra if needed…

Identity Matrix and Order Calculate each of the following. What do you notice?

[1 00 1 ][ 2 5

−3 4] [1 47 −2][2 2

3 5] [1 47 −2][2 2

3 5]a) b)i) ii)

¿ [ 2 5−3 4 ]

[1 00 1 ]

This is the ‘identity’ matrix for the 2x2 size

Multiplying another Matrix by it leaves the answer unchanged

It is the Matrix equivalent to multiplying by 1 in regular arithmetic

¿ [14 228 4 ] ¿ [16 4

38 2 ]You get different answers when you multiply Matrices in a different order

This is important as it is different to regular arithmetic where 2 x 3 = 3 x 2 etc

So ensure you always multiply in the order you are asked to!

(1×2 )+(0×−3)(1×5 )+(0×4 )(0×2 )+(1×−3)(0×5 )+(1×4 )

(1×2 )+(4×3) (1×2 )+(4×5)(7×2 )+(−2×3)(7×2 )+(−2×5)

(2×1 )+(2×7)(2×4 )+(2×−2)(3×1 )+(5×7 )(3×4 )+(5×−2)

Identity Matrix

Solving Matrix Problems

Solving Matrix problems

Solving Matrix problems

Matrix Algebra (5)You need to be able to find the

inverse of a Matrix

As you saw last lesson, the inverse of a Matrix is the Matrix you multiply it by to

get the Identity Matrix:

Remember that this is the Matrix equivalent of the number 1. Multiplying another 2x2 matrix by this will leave the

answer unchanged.

Also remember that from last lesson, the determinant of a matrix is given by:

[1 00 1 ]

𝑨=[𝑎 𝑏𝑐 𝑑 ]|𝑨|=𝑎𝑑−𝑏𝑐 for

Given: 𝑨=[𝑎 𝑏𝑐 𝑑 ]

𝑨−1= 1𝑎𝑑−𝑏𝑐 [ 𝑑 −𝑏

−𝑐 𝑎 ]

This means ‘the inverse of

A’

Remember this part is the ‘determinant’

Pay attention to how these

numbers have changed!

Matrix Algebra (5)You need to be able to find the

inverse of a Matrix

Find the inverse of the matrix given below:

𝑨=[𝑎 𝑏𝑐 𝑑 ] 𝑨−1= 1

𝑎𝑑−𝑏𝑐 [ 𝑑 −𝑏−𝑐 𝑎 ]

[3 24 3]

𝑨=[3 24 3 ]

𝑨−1=¿1

(3×3 )− (2×4 )[ 3 −2−4 3 ]

𝑨−1=¿11 [ 3 −2−4 3 ]

[ 3 −2−4 3 ]𝑨−1=¿

Replace the

numbers as

aboveWork

out the fraction

…

… which in this case you don’t need to write!

𝑆𝑜 : [3 24 3 ][ 3 −2

−4 3 ]=[1 00 1 ]

Matrix Transformation

Complete the sheet

Extr

a Q

uesti

ons