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Identity & Inverse Matrices

Identity & Inverse Matrices. Section 4-7. Identity Matrices. In the multiplication of numbers, the identity element is the number 1 since x ·1 = x for every value of x . (it gives the original number its identity back). For matrices, the number 1 is.

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Identity & Inverse Matrices

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  1. Identity & Inverse Matrices Section 4-7

  2. Identity Matrices In the multiplication of numbers, the identity element is the number 1 since x ·1 = x for every value of x. (it gives the original number its identity back) For matrices, the number 1 is If you multiply the matrix I with any matrix P and the result is the matrix P, then I is known as the identity matrix.

  3. 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 = N.B. When referring to the multiplicative identity, it is usually called "the identity matrix".

  4. 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

  5. Matrix 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.

  6. Given A and the inverse of A, denoted by A-1 IMPT NOTE : if two matrices are inverses and you multiply them, then the result is the IDENTITY MATRIX.

  7. 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. Step 2 : The inverse of matrix A is

  8. Examples Find the inverse if it exists: Find the inverse if it exists: Impossible !!!

  9. Examples Determine whether each pair of matrices are inverses If 2 matrices are inverses, when you multiply them you get the identity matrix. Yes – they’re inverses!

  10. 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.

  11. Using Matrices to Solve Simultaneous Equations • When the simultaneous equations is expressed in the matrix form, and if the determinant of the 22 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.

  12. • Step 2 : Find determinant of • Step 3 : If , then • Step 3 : If , Using Matrices to Solve Simultaneous Equations • Step 1 : Given ax + by = h • and cx + dy = k the equations have no unique solution.

  13. Ex 9D Page 214 • Class work: • Q1, 3, 5,8 • Q10 • Q12 • Q13 • Q14 • Homework: • Q2, 4, 6 • Q9 • Q11

  14. 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.

  15. 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?

  16. 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".

  17. 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.

  18. B A E C D R = 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 the loop at B gives 2 routes from B to B but the loop at D gives only 1 route because it is one-way only.

  19. Multiplying this matrix by itself gives R2which gives the number of possible two-stage routes from place to place. E.g. the number in the 1st 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, R3gives the number of possible three-stage routes from place to place and vice versa.

  20. A spreadsheet can be used for the tedious matrix operations as shown below.

  21. 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 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. 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.

  22. (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 =

  23. 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.

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