Dr. Ameria Eldosoky Discrete mathematics Discrete Mathematical Structures: Theory and Applications

Matrices

Matrices

Matrices

Matrices

Matrices

Matrices

Matrices Two matrices are added only if they have the same number of rows and the same number of columns To determine the sum of two matrices, their corresponding elements are added

Matrices

Matrices

Matrices

Matrices

Matrices The multiplication AB of matrices A and B is defined only if the number of rows and columns of A is the same as the number of rows and of B

Matrices Figure 4.1 Let A = [aij]m×n be an m × n matrix and B = [bjk ]n×p be an n × p matrix. Then AB is defined To determine the (i, k)th element of AB, take the ith row of A and the kth column of B, multiply the corresponding elements, and add the result Multiply corresponding elements as in Figure 4.1

Matrices

Matrices The rows of A are the columns of AT and the columns of A are the rows of AT

Matrices Boolean (Zero-One) Matrices Matrices whose entries are 0 or 1 Allows for representation of matrices in a convenient way in computer memory and for design and implement algorithms to determine the transitive closure of a relation

Matrices Boolean (Zero-One) Matrices The set {0, 1} is a lattice under the usual “less than or equal to” relation, where for all a, b ∈ {0, 1}, a ∨ b = max{a, b} and a ∧ b = min{a, b}

Matrices

Matrices

Matrices