Discrete Structures – CNS2300 Text Discrete Mathematics and Its Applications Kenneth H. Rosen (5th Edition) Chapter 2 The Fundamentals: Algorithms, the Integers, and Matrices

Section 2.7 Matrices

Matrix Definition A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an mxn matrix. The plural of matrix is matrices. A matrix with the same number of rows as columns is called square. Two matrices are equal if they have the same number of rows and the same number of columns and the corresponding entries in every position are equal.

Examples 3x2 2x3 2x2 1x5

Sum of two matrices Let A = [aij] and B = [bij] be mxn matrices. The sum of A and B, is the mxn matrix that has aij + bij as its (i,j)th element. In other words, A + B = [aij + bij].

Examples

Examples Not Possible

Matrix Multiplication Let A be an mxk matrix and B be a kxn matrix. The product of A and B, denoted by AB, is the mxn matrix with (i,j)th entry equal to the sum of the products of the corresponding elements from the ith row of A and the jth column of B. In other words, if AB = [cij], then

Example 3x3 3x2 2x3

Example 3x3 3x2 2x3

Example 3x2 2x3 3x3

Example 3x2 2x3 3x3

Algorithm procedure matrix multiplication(A,B:matrices) for i:=1 to m begin for j:=1 to n begin cij := 0 for q := 1 to k cij := cij + aiqbqj end end

Identity Matrix Exist as square matrices. AI = IA = A

Transpose of a matrix The transpose of a square matrix A is labeled AT and is found by interchanging rows and columns.

Symmetric Matrices A square matrix A is called symmetric if A = AT.

Zero-one Matrices

Join of two zero-one matrices

Join

Meet of two zero-one matrices

Boolean Product

Algorithm procedure matrix booleanProduct(A,B:matrices) for i:=1 to m begin for j:=1 to n begin cij := 0 for q := 1 to k cij := cij || aiq&&bqj end end

Boolean Product =

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