Introduction • Matrices have many important uses. • One of the simplest uses of matrices is to store data. • In fact, this is the most common use outside mathematics. • Matrices are used to solve linear systems and represent geometric figures and transformations. Building on this geometric interpretation, matrices can also be applied in the study of trigonometry. •Matrices are a powerful unifying concept, connecting ideas in mathematics to those in other content domains as well as connecting various branches of mathematics. As such, matrices can be a means of integrated whole rather than a set of isolated topics. •Matrices are a rich topic and that many uses of matrices are minor extensions or applications of content already in our syllabus. There are several reasons for introducing matrices into the syllabus: 1 • Matrices introduce a fundamental concept in discrete mathematics. • They model numerous realistic applications. • They furnish opportunities for doing arithmetic and algebraic computation in a new context. • Their properties and their operations lead to important theoretical results. Why Learn Matrices • Storing and organising numerical data • Operations with matrices • The identity Matrices • Inverse Matrices • Operations using a Spreadsheet • Some interesting Applications of Matrices ( not in syllabus) Storing and organising numerical data 6 rows and 7 columns of data, we say the matrix has size (order) 6 7 Example : Type Friction Textbooks General Reference Malay 25 47 22 30 Chinese 40 72 38 40 English 80 85 67 54 Matrix Type Malay Chinese English Friction 25 40 80 Textbook 47 72 85 General 22 38 67 Reference 30 40 54 Matrix Text Book Page 198 • Can you name the orders of the following matrices? (a) 2 by 2 (e) 3 by 5 (b) 3 by 1 (f) 2 by 1 (c) 1 by 4 (g) 1 by 1 (d) 3 by 3 Operations with Matrices • Addition and Subtraction of Matrices What do you notice about the order of the two matrices involved in each case? Rules for Matrix Addition and Matrix Subtraction • If P and Q are two matrixes of the same order, then the sum of P and Q, denoted by ( P + Q ), is a matrix with its elements being the sum of the corresponding elements of P and Q. • The difference of P and Q is denoted by (P–Q). • If P and Q are of different orders, then their sum and difference are not defined. Example : Ex 9A Page 201 • Class Work : • Q1 a to j Multiplication of a Matrix by a Real Number • Examples : Some Special Matrices • Square Matrix * A matrix has the same number of rows and columns. Examples : •Zero matrix or null matrix , denoted by O * Every element of a matrix is zero. * May be of any order Examples : Ex 9A Page 201 • Class work : • Homework : • Q3 • Q2 • Q4a, c, e, f • Q4b, d • Q5a, c, e • Q5b, d • Q6a • Q6b Operations With Matrices Matrix multiplication was invented in the 19th Century as a useful way of combining two suitable matrices. It multiplies elements in a row with elements in a column to produce one single number. To make up some matrix multiplication problems for yourself. And to investigate the condition on the sizes of two matrices that is necessary before you can be multiplied. Example : Maybeline bought 3 apples, 2 oranges and 4 pears.The cost of each item is 25 cents, $1 and 40 cents respectively. Find the total cost of the purchase. Answer : The total cost of the purchase = 3 25 cents + 2 100 cents + 4 40 cents = 435 cents = $ 4.35 Using Matrices : • Page 202, 203 • For example, • A 33 matrix multiplied by a 32 matrix will result in a 32 matrix. • A 32 matrix multiplied by a 23 matrix will result in a 33 matrix. • A 32 matrix multiplied by a 33 matrix is not possible. • In general : A m n matrix can be multiplied to a n p matrix to produce a m p matrix. Ex 9B Page 206 • Class Work : • Q1b, d, f • Homework : • Q2 • Q1a, c, e • Q3a • Q3b, c • Q4 • Q5 • Q6a, c, f • Q6b, d, e • Q8 • Q9 • Q10 • Q13 • Q11 • Q12 • Q14 Matrices Multiplication The prices of the drinks per cup are 55¢ for tea, 60¢ for coffee and 75¢ for drinking chocolate. (a) Form a matrix of prices and use it to find the total amount taken on each of the 3 days. (b) What information would be found by pre- multiplying V by (1 1 1)? explain why the matrix of prices can be a 31 or 13 matrix in (a) and what information is found if V is post-multiplied by Commutative and associative properties matrix multiplication is not commutative but associative. Explain using a problem Commutative and associative properties Prices A B C (AB)C is calculated by first finding the number of doors and information does the matrix AB What windows in each of the developments and gives? BA? then finding the total cost of windows and doors for each development. A(BC) is calculated by first finding the cost of the Calculate (AB)C and A(BC). doors and windows in each model and then finding their Explain why they are equal. total cost for each development. Commutative and associative properties • In general, matrix multiplication is not commutative, i.e. AB BA. ( Ex 9B Q 2 ) • However, matrix multiplication is associative. (AB)C = A(BC) ( Eg of previous 2 slides) • See example 4 on page 205 Identity Matrices • In the addition of numbers, the identity is 0 since x + 0 = x for every value of x. • In the multiplication of numbers, the identity is 1 since x 1 = x for every value of x. If you combine the matrix I with any matrix P and the result is the matrix P, then I is known as the identity matrix. Additive Identity : Is there a 2 2 identity matrix for matrix addition. i.e. •A + O = O + A = A , where O is the null matrix. For example, O = Multiplicative Identity : Is there a 2 2 identity matrix for matrix multiplication. i.e. A I = I A = A , where I is the identity matrix. For example, I = When referring to the multiplicative identity, it N.B. is usually called "the identity matrix". Characteristics of Identity matrix : • Is is a square matrix • All elements in the leading diagonal are 1. • All the other elements are 0. • Eg What do you obtain when A is multiplied by the identity matrix? AI = A or IA = A Inverse Matrices • The inverse of anything is that which will combine with it to give the identity. Is there an additive inverse for ? The additive inverse of is Reason : The additive inverse can be found easily. Multiplicative Inverse ** When we say "the inverse of a matrix", it is referring to the multiplicative inverse If A and B are two matrices and AB = BA = I, then A is said to be the inverse of B, denoted by B -1; B is said to be the inverse of A, denoted by A -1. IMPT NOTE : You are doing Matrices . Given A and the inverse of A, denoted by A -1 IMPT NOTE : You are doing Matrices . To find the inverse of a matrix A = . Step 1 : Find the determinant of the matrix A, denoted by det A Note : • If det A = 0, then the inverse of A is not defined. •Hence A does not have an inverse. •When a matrix does not possess an inverse, it is known as a singular matrix. Step 2 : The inverse of matrix A is Ex 9C Page 211 •Class Work: •Homework: •Q1a, b, h •Q1e,i •Q2a, d,e •Q2g,h •Q4 •Q3 •Q5 •Q7 •Q6 •Q8 •Q9 •Q10 Time to think • Can a matrix have more than one inverse? How to prove it? • Text Book Page 214 • 9.9 Some interesting Properties of Matrices • Q1, 2 Using Matrices to Solve Simultaneous Equations • To solve simultaneous equations by using simple algebra, if there is no solution or infinite solutions, what will you say about the two equations? • The simultaneous equations will represent either two parallel lines or the same straight line. Using Matrices to Solve Simultaneous Equations • When the simultaneous equations is expressed in the matrix form, and if the determinant of the 22 matrix is zero, then the two simultaneous equations will represent either two parallel lines or the same straight line. • The equations have no unique solution. Using Matrices to Solve Simultaneous Equations •Step 1 : Given ax + by = h and cx + dy = k •Step 2 : Find determinant of •Step 3 : If , then •Step 3 : If , the equations have no unique solution. Ex 9D Page 214 • Class work: • Homework: • Q1, 3, 5,8 • Q2, 4, 6 • Q10 • Q9 • Q12 • Q11 • Q13 • Q14 Why learn Matrices ? The interior design company is given the job of putting up the curtains for the windows, sliding doors and the living room of the entire new apartment block of the NTUC executive condominium. There are a total of 156 three- bedroom units and each unit has 5 windows, 3 sliding doors and 2 living rooms. Each window requires 6 m of fabric, each sliding door requires 14 m of fabric and each living room requires 22 m of fabric. Given that each metre of the fabric for the window cost $12.30, the fabric for the sliding door costs $14.50 per metre and each metre of the fabric for the living room is $16.50. We can write down three matrices whose product shows the total amount of needed to put up the curtains for each unit of the executive condominium. NE Message: The property market in Singapore went up very rapidly in the 1990’s. Many Singaporeans dream of owning a private property were dashed and many call for some form of help from the government to realise their dream. NTUC Choice Home was set up to go into property business as a way of stabilising the market and to help Singaporeans achieve their dream of owning private properties. With the onset of the Asian economic crises, the property market went under and the public start to question the need for NTUC Choice Home and urged NTUC to dissolve NTUC Choice Homes. Do you think this is a good request? How long do you think it will take to set up a company to run the property business? Operations using a Spreadsheet The Microsoft Excel matrix functions are: • MDETERM(array) Returns the matrix determinant of an array • MINVERSE(array) Returns the inverse of the matrix of an array • MMULT(array A, array B) Returns the matrix product • TRANSPOSE(array) Returns the transpose of an array. The first row of the input becomes the first column of the output array, etc. • *Except for MDETERM(), these are array functions and must be completed with "Crtl+shift+Enter". Some Interesting Applications • Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs. The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points. If the edges are assigned a direction, the graph is called directed. • Cryptography Matrices are also used in cryptography, the art of writing or deciphering secret codes. Routes Matrices Example If 5 places A, B, C, D, E are connected by a road system shown in the graph. The arrows denote one-way roads, then this can be listed as B A E the loop at B gives 2 routes from B to B but the loop at D gives C only 1 route because it is one-way only. R= D • Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place. E.g. the number in the 1 st row, 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA, another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road.) • Similarly, R3 gives the number of possible three- stage routes from place to place and vice versa. A spreadsheet can be used for the tedious matrix operations as shown below. Cryptography • One way of encoding is associating numbers with the letters of the alphabet as show below. This association is a one-to-one correspondence so that no possible ambiguities can arise. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 In this code, the word PEACE looks like 11 22 26 24 22. Suppose we want to encode the message: MATHS IS FUN If we decide to divide the message into pairs of letters, the message becomes MA TH IS SF UN. • (If there is a letter left over, we arbitrarily assign Z to the last position). Using the correspondence of letters to numbers given above, and writing each pair of letters as a column vector, we obtain • Choose an arbitrary 2 2 matrix A which has an inverse A-1. Say A = and A-1 = Now transform the column vectors by multiplying each of them on the left by A: The encoded message is 106 66 71 45 70 44 79 50 51 32. To decode, multiple by A-1 and reassigning letters to the numbers.