Revision on Matrices

Finding the order of, Addition, Subtraction and the Inverse of

Matices

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

several days. Summarise the information into a single table.

You can summarise the table by adding corresponding

columns together!

TOTAL TVs Radios PhonesDAY 1 15 7 26DAY 2 9 8 18DAY 3 16 7 20DAY 4 22 9 23

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

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 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

PlenaryCalculate the values of x and y in the matrix

equation below.

[2𝑥 43 𝑥 ]+[3 𝑦 7

2 − 𝑦 ]=[5 115 5 ]

2 𝑥+3 𝑦=5𝑥− 𝑦=5

1)

2)

2 𝑥+3 𝑦=51)

3 𝑥−3 𝑦=152)

5 𝑥=20𝑥=4

Multiply all by 3

Add 1) and 2)

Divide by 5

𝑦=−1You can then find y by

substitution!

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

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!

b) Calculate the following:

[−3 0 1 2 ][ 4−215 ] Multiply each number in the row with the

corresponding number in the column

(−3×4 )+(0×−2 )+(1×1 )¿−1

+(2×5 )

Show workings like these – it is essential to to have a good routine in place when we move onto bigger

Matrices!

PlenaryThe 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 (3)Multiplying Matrices together

Lets have a quick reminder of last lesson!

Remember you multiply the terms in the row by their corresponding terms in the column

Then we calculate the sum of these multiplications

a) Calculate the value of the following:

[ 4 6 −1 ][374 ](4×3 )+(6×7 )+(−1×4 )

¿50

Matrix Algebra (3)Multiplying Matrices together

Matrices can only be multiplied if the number of columns in the first is the same as the number of rows in the second.

[6 5 −2 ][558 ]1 x 3 3 x 1

These numbers

have to be the same!

These numbers give the dimensions of the final matrix!

1 x 1

¿ [39 ] [ 3 22 56 −1] ¿ [ 22 9 29 −6

22 17 45 734 3 23 −27 ]

3 x 2 2 x 4

These numbers

have to be the same!

These numbers give the dimensions of the final matrix!

3 x 4

[6 12 3

5 −47 3 ]

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:

[154 ][ 4 3 ]

3 x 1 1 x 2 3 x 2¿ [¿ ]

(1×4 ) (1×3)

¿ [ 4 320 1516 12]

(5×4 ) (5×3)(4×4 ) (4×3)

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)

Plenary• Why do we multiply matrices like this? Matrices were originally developed as a method to solve and rearrange multiple linear equations and expressions

𝑝=13𝑢−20𝑣𝑞=2𝑢+6 𝑣

𝑢=3 𝑥+7 𝑦𝑣=−2 𝑥+11 𝑦

Here we have two pairs of equations

Write p and q in terms of x and y

Substitute the first equations into the second…

𝑝=13 (3 𝑥+7 𝑦 )−20(−2 𝑥+11 𝑦)𝑞=2(3𝑥+7 𝑦)+6(−2𝑥+11 𝑦 )

𝑝=79 𝑥−129 𝑦𝑞=−6 𝑥+70 𝑦

Replace the u terms and the v terms

Multiply out and simplify

𝑝=13𝑢−20𝑣𝑞=2𝑢+6 𝑣

𝑢=3 𝑥+7 𝑦𝑣=−2 𝑥+11 𝑦

Mathematicians realised that for more complicated equations, they needed a more efficient method…

They wrote the sets of equations as matrices and multiplied them using the method you have seen!

[ 3 7−2 11][13 −20

2 6 ](13×3 )+(−20×−2) (13×7 )+(−20×11)(2×3 )+(6×−2) (2×7 )+(6×11)

¿ [79 −129−6 70 ]

This method then stuck and is the way matrix multiplication has been defined ever since!

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 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 ]

[6 −54 −2]

𝑨=[6 −54 −2]

𝑨−1=¿1

(6×−2 )− (4×−5 )[−2 −54 6 ]

𝑨−1=¿18 [−2 −54 6 ]

Replace the

numbers as

aboveWork

out the fraction

…

𝑆𝑜 : [6 −54 −2] [− 28 − 58

48

68 ]=[1 0

0 1]

You can include the fractional part in the Matrix

Obviously you would simplify the fractions if you did!

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

inverse of a Matrix

It is important to note that not every Matrix actually has an inverse!

𝑨=[𝑎 𝑏𝑐 𝑑 ]

𝑨−1= 1𝑎𝑑−𝑏𝑐 [ 𝑑 −𝑏

−𝑐 𝑎 ]

If this calculation is equal to 0, the Matrix does not have an inverse

The reason is that we are not able to divide by 0!

PlenaryCalculate the values of a, b, c and d in the calculation below using

Simultaneous equations.

[ 4 −8−3 6 ] [𝑎 𝑏

𝑐 𝑑 ]=[1 00 1]

𝑨=[𝑎 𝑏𝑐 𝑑 ] 𝑨−1= 1

𝑎𝑑−𝑏𝑐 [ 𝑑 −𝑏−𝑐 𝑎 ]

(4×𝑎)+(−8×𝑐)(4×𝑏 )+(−8×𝑑)(−3×𝑎 )+(6×𝑐)(−3×𝑏)+(6×𝑑)

4𝑎−8𝑐−3 𝑎+6𝑐

4𝑏−8𝑑−3 𝑏+6𝑑

4𝑎−8𝑐=1−3 𝑎+6𝑐=0

12𝑎−24𝑐=3−12𝑎+24 𝑐=0

Comparing the algebraic

versions to the answer above…

However you try to eliminate a or c, the other will be eliminated too so the equations are not solvable

The implication is that the Matrix above has no inverse

You will see that if you calculated the determinant, it is equal to 0!

x3

x4

Click on this link to go and practice some matrix questions, you will need pen and paper then the solutions will appear;

http://igcse.at.ua/IGCSE-MATHS/IGCSE20Matrices.pdf

This one is multiple choice, it s says you are only allowed 3 questions, if you then press the back button you can complete the other 3 matrix questions – there is only 6 altogether.

http://math-quiz.co.uk/gcse-maths/matrices

Website and videos• This website has lots of videos and notes –

make some of your own notes in your book, get some earphones and listen to some videos.

• http://www.onlinemathlearning.com/matrices-lessons.html

• Or this link goes through matrix multiplication again;

• https://www.youtube.com/watch?v=OAh573i_qn8#t=97

The next step;• The next step with matrices that you will have

to do is to describe a rotation, reflection, enlargement or shifting of a shape by a matrix. This website has lots of good notes you can make – learn the matrices for each it will make your life a lot easier;

• http://www.mathelaureate.com/wp-content/uploads/2013/08/IGCS-Transformation.pdf

• Note; you DO NOT have to do shears or stretches

A big file with lots more notes;

If you are really getting stuck into the transformation matrices read on!!

http://www.cimt.plymouth.ac.uk/projects/mepres/alevel/fpure_ch9.pdf

Extension;

watch this video of an exam question being completed;

https://www.youtube.com/watch?v=qD1ZWXO5bHc

Exam solutions have quite a few videos that can be helpful.