Chapter-1_Basic Structure-MATRICES.pdf

DISCRETE MATHEMATICS I for SE Engr. Marian Mie Alimo-ot EMath 1105 Engr. Yeseil Sacramento EMath 1105 – DISCRETE MATHEMATICS I for SE CHAPTER 1: BASIC STRUCTURES...

DISCRETE MATHEMATICS I for SE Engr. Marian Mie Alimo-ot EMath 1105 Engr. Yeseil Sacramento EMath 1105 – DISCRETE MATHEMATICS I for SE CHAPTER 1: BASIC STRUCTURES SETS FUNCTIONS SEQUENCES AND SUMMATIONS MATRICES EMath 1105 – DISCRETE MATHEMATICS I for SE MATRICES EMath 1105 – DISCRETE MATHEMATICS I for SE MATRICES 𝑎!" − 𝑟𝑜𝑤 2, 𝑐𝑜𝑙𝑢𝑚𝑛 3 EMath 1105 – DISCRETE MATHEMATICS I for SE MATRIX ARITHMETIC - SUM The sum of two matrices of the same size is obtained by adding elements in the corresponding positions. Matrices of different sizes cannot be added, because the sum of two matrices is defined only when both matrices have the same number of rows and the same number of columns. EMath 1105 – DISCRETE MATHEMATICS I for SE MATRIX ARITHMETIC – SUM EMath 1105 – DISCRETE MATHEMATICS I for SE MATRIX ARITHMETIC - PRODUCT Ø A product of two matrices is defined only when the number of columns in the first matrix equals the number of rows of the second matrix. Ø The product of two matrices is not defined when the number of columns in the first matrix and the number of rows in the second matrix are not the same. EMath 1105 – DISCRETE MATHEMATICS I for SE MATRIX ARITHMETIC - PRODUCT EMath 1105 – DISCRETE MATHEMATICS I for SE MATRIX ARITHMETIC - PRODUCT EMath 1105 – DISCRETE MATHEMATICS I for SE TRANSPOSES and POWERS OF MATRICES EMath 1105 – DISCRETE MATHEMATICS I for SE TRANSPOSES and POWERS OF MATRICES Ø Multiplying a matrix by an appropriately sized identity matrix does not change this matrix. Ø In other words, when A is an m×n matrix, we have EMath 1105 – DISCRETE MATHEMATICS I for SE TRANSPOSES and POWERS OF MATRICES ! 2 1 Example: A= −1 0 2 1 2 1 2 1 = % % −1 0 −1 0 −1 0 2 1 3 2 = % −1 0 −2 −1 4 3 = −3 −2 EMath 1105 – DISCRETE MATHEMATICS I for SE TRANSPOSES and POWERS OF MATRICES EMath 1105 – DISCRETE MATHEMATICS I for SE TRANSPOSES and POWERS OF MATRICES Ø A matrix is symmetric if and only if it is a square matrix and it is symmetric with respect to its main diagonal (which consists of entries that are in the ith row and ith column for some i). EMath 1105 – DISCRETE MATHEMATICS I for SE TRANSPOSES and POWERS OF MATRICES Which of the following matrices is symmetric? 1 1 −2 1 3 A= 1 1 B= 1 0 −1 1 1 3 −1 2 3 0 1 C= 0 2 −1 1 1 −2 EMath 1105 – DISCRETE MATHEMATICS I for SE ZERO-ONE MATRICES Ø A matrix all of whose entries are either 0 or 1 is called a zero–one matrix. Ø Algorithms using these structures are based on Boolean arithmetic with zero–one matrices. Ø This arithmetic is based on the Boolean operations ∧ (meet/AND) and ∨ (join/OR), which operate on pairs of bits, defined by EMath 1105 – DISCRETE MATHEMATICS I for SE ZERO-ONE MATRICES EMath 1105 – DISCRETE MATHEMATICS I for SE ZERO-ONE MATRICES EMath 1105 – DISCRETE MATHEMATICS I for SE ZERO-ONE MATRICES EMath 1105 – DISCRETE MATHEMATICS I for SE ZERO-ONE MATRICES Ø The Boolean product of A and B is obtained in an analogous way to the ordinary product of these matrices, but with addition replaced with the operation ∨ and with multiplication replaced with the operation ∧. EMath 1105 – DISCRETE MATHEMATICS I for SE ZERO-ONE MATRICES EMath 1105 – DISCRETE MATHEMATICS I for SE ZERO-ONE MATRICES EMath 1105 – DISCRETE MATHEMATICS I for SE ZERO-ONE MATRICES EMath 1105 – DISCRETE MATHEMATICS I for SE ZERO-ONE MATRICES Exercise. Perform the given operations below. EMath 1105 – DISCRETE MATHEMATICS I for SE ZERO-ONE MATRICES Answers: EMath 1105 – DISCRETE MATHEMATICS I for SE Inverse of a 2 x 2 Matrix Exchange elements of main diagonal Change sign in elements off main diagonal Divide resulting matrix by the determinant éa b ù A=ê ú ë c d û EMath 1105 – DISCRETE MATHEMATICS I for SE Inverse of a 2 x 2 Matrix Example EMath 1105 – DISCRETE MATHEMATICS I for SE Practice Exercises 1. Find A + B for each of the following. EMath 1105 – DISCRETE MATHEMATICS I for SE Practice Exercises 2. Find AB for each of the following. EMath 1105 – DISCRETE MATHEMATICS I for SE Practice Exercises 3. Find a matrix A that would satisfy the equation. EMath 1105 – DISCRETE MATHEMATICS I for SE REFERENCE: Rosen, K. H. (2012). Discrete Mathematics and Its Applications, 8th Edition: McGrawHill