revision on matrices finding the order of, addition, subtraction and the inverse of matices
DESCRIPTION
Starter These 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 βMatrix Algebraβ 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 ATVsRadiosPhones DAY DAY 2628 DAY 3729 DAY Shop BTVsRadiosPhones DAY DAY DAY DAY 4125 We can use matrices to represent the information aboveβ¦TRANSCRIPT
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 βMatrix Algebraβ
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]ΒΏ [15 7 26
9 8 1816 7 2022 9 23 ]
We can use matrices to represent the information aboveβ¦
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 βorderβ 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 β11 3 ] b) [1 0 2 ]
d)[ 4β1] [ 3 2β1 10 β3]
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 βorderβ of
a matrix (its dimensions)
You can add and subtract matrices only when they have
the same dimensions
π¨=[ 5 7 4β6 β2 3 ]
π©=[ 8 β2 0β3 8 β1]
Calculate A + B
[ 5 7 4β6 β2 3 ]+[ 8 β2 0
β3 8 β1]
ΒΏ [ΒΏ ]26β94513
Calculate A - B
[ 5 7 4β6 β2 3 ]β[ 8 β2 0
β3 8 β1 ]
ΒΏ [ΒΏ ]4β10β349β3
PlenaryCalculate the values of x and y in the matrix
equation below.
[2π₯ 43 π₯ ]+[3 π¦ 7
2 β π¦ ]=[5 115 5 ]
2 π₯+3 π¦=5π₯β π¦=5
1)
2)
2 π₯+3 π¦=51)
3 π₯β3 π¦=152)
5 π₯=20π₯=4
Multiply all by 3
Add 1) and 2)
Divide by 5
π¦=β1You 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
π¨=[ 5 2β4 0]
π¨=[ 5 2β4 0]a)
2 π¨=[ 10 4β8 0 ]
π¨=[ 5 2β4 0]b)
β3 π¨=[β15 β612 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
(2Γ4 )+(5Γ6 )+(3Γ1 )ΒΏ 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:
[β3 0 1 2 ][ 4β215 ] Multiply each number in the row with the
corresponding number in the column
(β3Γ4 )+(0Γβ2 )+(1Γ1 )ΒΏβ1
+(2Γ5 )
Show workings like these β 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!
[ π₯ π¦ ] [53]=[20 ]
[ π¦ β2 ] [ 2π₯ ]=[β24 ]
5 π₯+3 π¦=20
2 π¦β2 π₯=β24
10 π₯+6 π¦=40
10 π¦β10 π₯=β120
16 π¦=β80π¦=β5π₯=7
As an equation
As an equation Multiply by
2
Multiply by 5
Add the two equations together
Divide by 16
Then find x
[ π₯ π¦ ] [53]=[20 ] [ π¦ β2 ] [2π₯ ]=[β24 ]
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 β1 ][374 ](4Γ3 )+(6Γ7 )+(β1Γ4 )
ΒΏ50
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 β2 ][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
ΒΏ [39 ] [ 3 22 56 β1] ΒΏ [ 22 9 29 β6
22 17 45 734 3 23 β27 ]
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 β47 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 firstβ¦)
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 β 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
ΒΏ [ΒΏ ]
(5Γ3 )+(6Γ4)(5Γ1 )+(6Γ2)
ΒΏ [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 firstβ¦)
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 β 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 2ΒΏ [ΒΏ ]
(1Γ4 ) (1Γ3)
ΒΏ [ 4 320 1516 12]
(5Γ4 ) (5Γ3)(4Γ4 ) (4Γ3)
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 firstβ¦)
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 β it will already be set out in the correct position!
Lets see an example!
Calculate the following:
[ 1 3β5 0 ][3 7
4 β1]2 x 2 2 x 2 2 x 2
ΒΏ [ΒΏ ]
(1Γ3 )+(3Γ4 )
ΒΏ [ 15 4β15 β35 ]
(1Γ7 )+(3Γβ1)(β5Γ3 )+(0Γ4)(β5Γ7 )+(0Γβ1)
Plenaryβ’ Why do we multiply matrices like this? Matrices were originally developed as a method to solve and rearrange multiple linear equations and expressions
π=13π’β20π£π=2π’+6 π£
π’=3 π₯+7 π¦π£=β2 π₯+11 π¦
Here we have two pairs of equations
Write p and q in terms of x and y
Substitute the first equations into the secondβ¦
π=13 (3 π₯+7 π¦ )β20(β2 π₯+11 π¦)π=2(3π₯+7 π¦)+6(β2π₯+11 π¦ )
π=79 π₯β129 π¦π=β6 π₯+70 π¦
Replace the u terms and the v terms
Multiply out and simplify
π=13π’β20π£π=2π’+6 π£
π’=3 π₯+7 π¦π£=β2 π₯+11 π¦
Mathematicians realised that for more complicated equations, they needed a more efficient methodβ¦
They wrote the sets of equations as matrices and multiplied them using the method you have seen!
[ 3 7β2 11][13 β20
2 6 ](13Γ3 )+(β20Γβ2) (13Γ7 )+(β20Γ11)(2Γ3 )+(6Γβ2) (2Γ7 )+(6Γ11)
ΒΏ [79 β129β6 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 ]
π¨=[π ππ π ]|π¨|=ππβππ for
Given: π¨=[π ππ π ]
π¨β1= 1ππβππ [ π βπ
βπ π ]
This means βthe inverse of
Aβ
Remember this part is the βdeterminantβ
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:
π¨=[π ππ π ] π¨β1= 1
ππβππ [ π βπβπ π ]
[3 24 3]
π¨=[3 24 3 ]
π¨β1=ΒΏ1
(3Γ3 )β (2Γ4 )[ 3 β2β4 3 ]
π¨β1=ΒΏ11 [ 3 β2β4 3 ]
[ 3 β2β4 3 ]π¨β1=ΒΏ
Replace the
numbers as
aboveWork
out the fraction
β¦
β¦ which in this case you donβt need to write!
ππ : [3 24 3 ][ 3 β2
β4 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:
π¨=[π ππ π ] π¨β1= 1
ππβππ [ π βπβπ π ]
[6 β54 β2]
π¨=[6 β54 β2]
π¨β1=ΒΏ1
(6Γβ2 )β (4Γβ5 )[β2 β54 6 ]
π¨β1=ΒΏ18 [β2 β54 6 ]
Replace the
numbers as
aboveWork
out the fraction
β¦
ππ : [6 β54 β2] [β 28 β 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!
π¨=[π ππ π ]
π¨β1= 1ππβππ [ π βπ
βπ π ]
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 β8β3 6 ] [π π
π π ]=[1 00 1]
π¨=[π ππ π ] π¨β1= 1
ππβππ [ π βπβπ π ]
(4Γπ)+(β8Γπ)(4Γπ )+(β8Γπ)(β3Γπ )+(6Γπ)(β3Γπ)+(6Γπ)
4πβ8πβ3 π+6π
4πβ8πβ3 π+6π
4πβ8π=1β3 π+6π=0
12πβ24π=3β12π+24 π=0
Comparing the algebraic
versions to the answer aboveβ¦
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 β there is only 6 altogether.
http://math-quiz.co.uk/gcse-maths/matrices
Website and videosβ’ This website has lots of videos and notes β
make some of your own notes in your book, get some earphones and listen to some videos.
β’ http://www.onlinemathlearning.com/matrices-lessons.html
β’ Or this link goes through matrix multiplication again;
β’ https://www.youtube.com/watch?v=OAh573i_qn8#t=97
The next step;β’ 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 β learn the matrices for each it will make your life a lot easier;
β’ http://www.mathelaureate.com/wp-content/uploads/2013/08/IGCS-Transformation.pdf
β’ 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.