1. CS 285- Discrete Mathematics Lecture 11

2. Section 3.8 Matrices Introduction Matrix Arithmetic Transposes and Power of Matrices Zero – One Matrices Boolean Product of Matrices 2 Matrices

3. Introduction Definition: a matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m x n matrix. A matrix with the same number of rows and columns is a square matrix Two matrices are equal if they have the same number of rows and columns and the corresponding entries in every position are equal. Matrices 3

4. Matrix Arithmetic – Addition Let A = [ a ij ] and B = [b ij ] be mxn matrices. The sum of A and B, denoted by A + B, is the m x n matrix that has a ij +b ij as its (i,j)th element. In other words, A + B = [ a ij + b ij ] Ex. Matrices 4

5. Matrix Arithmetic - Product Let A be an m x k matrix and B be a k x n matrix. The product of A and B, denoted by AB, is the m x n matrix with its (i,j)th entry equal to the sum of products of the corresponding elements from the ith row of A and the jth column of B. AB = [c ij ], c ij = a i1 b 1j + a i2 b 2j + ….. + a ik b kj Ex. Matrices 5

6. Product –Example II Matrices 6

7. Identity Matrix Definition: The identity matrix of order n is the n x n matrix I n = [ δ ij ], where δ ij = 1 if i = j and δ ij = o if i ≠ j. Hence: I n = A I n = I m A = A, where A is an m x n matrix Matrices 7

8. Powers of Matrices The powers of matrices is the product of this matrix by it self for r times. Let A be an m x n matrix A 0 = I n, A r = AAA … A Matrices 8 r times

9. Transposes Definition: The transpose of the mxn matrix A, denoted by A t, is the n x m matrix obtained by interchanging the rows and the columns of A. If A t = [ b ij ], then b ij = a ji for i = 1,2, …,n and j = 1,2, …, m Ex. The transpose of matrix A is Matrices 9 t

10. Symmetric Matrices Definition: A square matrix A is called symmetric if A = A t. Thus A = [ a ij ] is symmetric of a ij = a ji for all i and j with 1 ≤ i ≤ n and 1 ≤ j ≤ m Ex. Matrices 10

11. Zero-One Matrices Definition: A matrix with entries that are either 0 or 1. Let A = [ a ij ] and B = [ b ij ] be m x n zero-one matrices. Join of A and B is a zer0-one matrix with the (i,j)th entry a ji b ij. A B Meet of A and B is a zer0-one matrix with the (i,j)th entry a ji b ij. A B Matrices 11

12. Examples Matrices 12

13. Matrices 13