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# The sum of a and b denoted by a b is the m n matrix

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Unformatted text preview: matrices used to build models of: ༉  Transportation systems. ༉  Communication networks. ༉  Algorithms based on matrix models will be presented in later chapters. Here we cover the aspect of matrix arithmetic. Matrix Deﬁnition: a matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m × n matrix. ༉  The plural of matrix is matrices. ༉  A matrix with the same number of rows as columns is called square matrix. ༉  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. 3 × 2 matrix Nota\$on ༉  Let m and n be positive integers and let ༉  The ith row of A is the 1 × n matrix [ai1, ai2, …, ain]. The jth column of A is the m × 1 matrix: ༉  The (i , j)th element or entry of A is the element aij. Matrix Addi\$on Deﬁntion: let A = [aij] and B = [bij] be m × n matrices. The sum of A and B, denoted by A + B, is the m × n matrix that has aij + bij as its (i , j)th element. In other words, A + B = [aij + bij]. Example: Note that matrices of different sizes can not be added. Matrix Mul\$plica\$on Deﬁnition: let A be an n × k matrix and B be a k × n matrix. The product of A and B, denoted by AB = [cij] , is the m × n matrix where cij = ai1b1j + ai2b2j + … + akjb2j . Example: The product of A and B is undeWined when the number of columns in the 1st matrix is not the same as the number o...
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