Chapter 12: Matrices II TOPIC The Inverse Matrix TEXT: Essential Further Mathematics DATE SET PAGE(S) WORK 421 Exercise 12A: Q1 4 Applying the Inverse Matrix: Solving Simultaneous Equations 428 Exercise 12B: Q1-11 Transition Matrices and their applications 430 - Exercise 12C1 12C4 Setting up and interpreting transition matrices 432 434 Exercise 12C1: Q1 Exercise 12C2: Q1-5 Using recursion to generate state matrices and steady state solutions 443 Exercise 12C3: Q1 5 Transition matrix modelling: Adding to and subtracting from 446 Exercise 12C4: Q1 2 Chapter Review Ex 12A The Inverse matrix 449 Multiple-choice questions: Q 1-22 Extended-response questions: Q1 The Identity Matrix A square matrix with all ones in the leading diagonal and zeros elsewhere is known as an identity matrix and is represented by the symbol I. Examples are given below: Identity matrices have the property such that, for a square matrix A: IA = AI = A Thus, it can be seen that an identify matrix has the same role in matrix arithmetic as the number one in normal arithmetic. i.e. just as (1 x 3 = 3 x 1 = 3) IA = AI = A 1
The Inverse Matrix (A -1 ) The inverse of a square matrix A, is called A -1 and has the property that: 1 1 A A A A I 2
The Determinant of a Matrix, det(a) OR A The determinant of a matrix is required in the evaluation of the inverse of that matrix The determinant of a matrix A is written as det (A) or A If then Example 1 Find the determinant of the matrices: A = 2 3 3 5 B = 2 4 3 3 C = 2 7 4 14 Finding the Inverse of a Matrix The Inverse of a 2 x 2 Matrix If, then its inverse A -1 is given by A -1 = provided that det (A) 0 1 det A d c b a Hence, the inverse of a matrix only exists if its determinant is not equal to zero. Example 2 (a) If, find A -1. (b) If, find B -1. 3
Example 3 It is given that (a) If the determinant of P is -3, find the value of a. (b) Hence, find the inverse of P. Example 4 If the matrix (a) find A. (b) is A -1 defined? NOTE: Matrix A is an example of a singular matrix; its inverse is undefined since its determinant is zero. 4
Using CAS to determine the determinant and inverses of n n matrices (where n 2) 5
Example 5 For the matrix B = (i) find B -3 0 2 2 1-1 1 4 2 (ii) find B -1 Example 6 For the matrix 2 0-3 -5 0 1 0 2 1 (i) find its determinant (ii) find its inverse 6
Applications of the inverse matrix:solving simultaneous linear equations Example 10 (a) Write a matrix equation that represents the pair of simultaneous equations: 4x + 2y = 5 3x + 2y = 2 (b) Solve this matrix equation. 7 NOTE: We say that this matrix equation has a unique solution.
Example 11 Solve the matrix equation 3 2 1 1 x = y 2 1 Example 12 Solve the matrix equation 1 0 0 0 1 1 3 0 1 x y z = 1 0 1 8
Simultaneous Equations that do not have a unique solution: All pairs of simultaneous equations can be graphed and their solution is their point of intersection written as a pair of coordinates. Sometimes simultaneous equations cannot be solved, meaning they don t have a unique solution. The reason that they cannot be solved is because not all square matrices have an inverse. A square matrix will not have an inverse if its determinant = 0. If the determinant = 0 (meaning the inverse is not defined) there are 2 types of graphs that are possible: 1. Equations are inconsistent: their graphs are parallel and thus do not intersect. Write these equations in matrix form and check the value of the determinant. 2. Equations are dependent: the graphs coincide (are the same graph) and thus do not intersect at a single point. Write these equations in matrix form and check the value of the determinant. 9
Example 13: Text Book Example 7 from Page 427 10
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Transition matrices Transition matrices are used to determine the likelihood of an event occurring, given what has occurred previously, and then they can allow you predict what is expected to occur in the future. Features of a transition matrix Columns add to 1 and Top heading of the columns is "From" Side heading of the rows is "To". 12
Example 20: Setting up a transition matrix A factory has a large number of machinery. Machines can be in one of two states, operating or broken. Broken machines are repaired and come back into operation, and vice-versa. On a given day: 85% of machines that are operational stay operating 15% of machines that are operating break down 5% of machines that are broken are repaired and start operating 95% of machines that are broken stay broken Construct a transition matrix to describe this situation. Use the columns to define the situation at the Start of the day and the rows to describe the situation at the End of the day. Example 21: Setting up a transition matrix The following information relates to a survey conducted in January 2010 on supermarket shopping. 12% of store A customers will shop at store B in the following month 36% of store A customers will shop at store C the following month 40% of store B customers will shop at store B in the following month 44% of store B customers will shop at store C the following month 14% of store C customers will shop at store A in the following month 7% of store C customers will shop at store B the following month Represent this information as a transition matrix. 13
Example 22 Complete the following transition matrices by filling in the missing elements. 0.85 0.15 0.3 0.05 0.55 0.65 0.15 0.68 0.12 0.03 0.97 14
Using recursion to generate state matrices step-by step. (d) The column matrix S 0 = 50 40 is called the initial state matrix. (i) Construct a new state matrix S1 to show the number of cars at each branch after 1 week. (ii) Construct a new state matrix S2 to show the number of cars at each branch after 2 weeks. (iii) Construct a new state matrix S3 to show the number of cars at each branch after 3 weeks. (iv) Construct a new state matrix S14 to show the number of cars at each branch after 14 weeks. (v) Construct a new state matrix S15 to show the number of cars at each branch after 15 weeks. 15
A rule for determining the state matrix of a system after n steps. Recall: A recurrence relation has two parts starting value and rule for finding the next term. The recurrence relation : is used to generate state matrices step-by-step, and relies on knowing the previous state matrix. It is a slow method. A more efficient method for finding any state matrix can be developed: SUMMARY: Two methods for finding the state matrix after n steps: If the previous state matrix is known: Finding a state matrix at any step: Important: What do the subscripts and superscripts add to? What do the subscripts and superscripts add to? 16
The Steady (or equilibrium) State solution. As the number of transitions increases indefinitely, the population being considered will approach a steady state or state of equilibrium. This means the number in each category of the state matrix will converge on a particular value and will stabilise as the number of times the event occurs increases. NOTE: In general, a simple rule for determining the state matrix after n steps is: Sn = T n S0 If Sn = Sn+1, then a steady state solution is arrived at. In general, we say that: If S0 is the initial state matrix, then the steady state matrix S, is given by: S = T n S0 as n tends to infinity. In practice, values of n around 15 to 30 will give a very close approximation to the steady state solution. 17
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Transition matrix modelling using the rule: If you want to increase (or decrease) the total number of objects in a system, at each step of the process, the following matrix recurrence relation must be used: NOTE: This recurrence rule does not lead to a simple rule for the state matrix after n steps. You have to work your way through this sort of problem step-by-step. 19
Skills Check Calculate the determinant of a matrix Know the properties of an inverse matrix Find the inverse of a square matrix using a calculator Use determinants to test a system of linear equations for solutions Use inverse matrices to solve systems of linear equations Use of the matrix recurrence relation:, to generate a sequence of state matrices, including an informal identification of the equilibrium or steady-state matrix in the case of regular state matrices Construct a transition matrix from a transition diagram and vice versa Construct a transition matrix to model the transitions in a population Use of the matrix recurrence relation to model systems that include external additions or reductions at each step of the process. 20