1. Revision on Matrices Finding the order of, Addition, Subtraction and the Inverse of Matices

2. Starter These 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! TOTALTVsRadiosPhones DAY 115726 DAY 29818 DAY 316720 DAY 422923 Shop ATVsRadiosPhones DAY 17312 DAY 2628 DAY 3729 DAY 410411 Shop BTVsRadiosPhones DAY 18414 DAY 23610 DAY 39511 DAY 4125

3. 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 ‘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 ATVsRadiosPhones DAY 17312 DAY 2628 DAY 3729 DAY 410411 Shop BTVsRadiosPhones DAY 18414 DAY 23610 DAY 39511 DAY 4125 We can use matrices to represent the information above…

4. Matrix Algebra To 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 b) d)  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

5. Matrix Algebra To 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 Calculate A + B Calculate A - B

6. Plenary Calculate the values of x and y in the matrix equation below. 1) 2) 1) 2) Multiply all by 3 Add 1) and 2) Divide by 5 You can then find y by substitution!

7. 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) b) 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

8. 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  Multiply each number in the row with the corresponding number in the column

9. 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:  Multiply each number in the row with the corresponding number in the column Show workings like these – it is essential to to have a good routine in place when we move onto bigger Matrices!

10. Plenary The values of x and y in these pairs of Matrices are the same. Calculate what x and y must be! As an equation Multiply by 2 Multiply by 5 Add the two equations together Divide by 16 Then find x

11. 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:

12. 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. 1 x 33 x 1 These numbers have to be the same! These numbers give the dimensions of the final matrix! 1 x 13 x 22 x 4 These numbers have to be the same! These numbers give the dimensions of the final matrix! 3 x 4

13. 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 x 22 x 21 x 2

14. 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: 3 x 11 x 2 3 x 2

15. 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: 2 x 2

16. Plenary Why do we multiply matrices like this?  Matrices were originally developed as a method to solve and rearrange multiple linear equations and expressions Here we have two pairs of equations Write p and q in terms of x and y  Substitute the first equations into the second… Replace the u terms and the v terms Multiply out and simplify 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! This method then stuck and is the way matrix multiplication has been defined ever since!

17. 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: for Given: This means ‘the inverse of A’ Remember this part is the ‘determinant’ Pay attention to how these numbers have changed!

18. Matrix Algebra (5) You need to be able to find the inverse of a Matrix Find the inverse of the matrix given below: Replace the numbers as above Work out the fraction … … which in this case you don’t need to write!

19. Matrix Algebra (5) You need to be able to find the inverse of a Matrix Find the inverse of the matrix given below: Replace the numbers as above Work out the fraction … You can include the fractional part in the Matrix  Obviously you would simplify the fractions if you did!

20. 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! 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!

21. Plenary Calculate the values of a, b, c and d in the calculation below using Simultaneous equations. 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

22. 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

23. 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

24. 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 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 https://www.youtube.com/watch?v=OAh573i _qn8#t=97

25. 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 http://www.mathelaureate.com/wp- content/uploads/2013/08/IGCS- Transformation.pdf Note; you DO NOT have to do shears or stretches

26. 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/mepr es/alevel/fpure_ch9.pdf

27. Extension; watch this video of an exam question being completed; https://www.youtube.com/watch?v=qD1ZWXO5 bHc Exam solutions have quite a few videos that can be helpful.