Lecture 2(1).pdf - OPRE 3333 Chapter 2 Larson Matrices Matrices • A matrix is a rectangular array of numbers(called elements arranged in orderly rows

Lecture 2(1).pdf - OPRE 3333 Chapter 2 Larson Matrices...

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OPRE 3333: Chapter 2 - Larson Matrices Matrices: A matrix is a rectangular array of numbers (called elements) arranged in orderly rows and columns. A = a 11 a 12 a 13 a 21 a 22 a 23 Each element is denoted as a ij , where subscripts denote the row ( i = 1 , . . . , m ) and column ( j = 1 , . . . , n ) location of the element. The index i is called the row subscript and index j is called the column subscript. Sizes of matrices: m × n Basic matrix operations: Addition and subtraction Multiplication Transpositions Matrix addition and subtraction: If A = [ a ij ] and B = [ b ij ] are matrices of the same size m × n , then their sum is the m × n matrix A + B = [ a ij + b ij ]. (Simply add (subtract for A - B ) elements from corresponding locations.) A 2 × 2 + B 2 × 2 = a 11 a 12 a 21 a 22 + b 11 b 12 b 21 b 22 = c 11 c 12 c 21 c 22 where c 11 = a 11 + b 11 c 12 = a 12 + b 12 c 21 = a 21 + b 21 c 22 = a 22 + b 22 A 2 × 2 - B 2 × 2 = a 11 a 12 a 21 a 22 - b 11 b 12 b 21 b 22 = c 11 c 12 c 21 c 22 where c 11 = a 11 - b 11 c 12 = a 12 - b 12 c 21 = a 21 - b 21 c 22 = a 22 - b 22 ** Note that the sum of two matrices of different sizes is undefined. Example: For A = 1 2 3 4 5 6 and B = 7 8 9 10 11 12 calculate C = A + B =
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Chapter 2 - OPRE 3333 2 C = A - B = Matrix scalar multiplication: If A = [ a ij ] is an m × n matrix and b is a scalar, then the scalar multiple of A by b is the m × n matrix bA = [ ba ij ]. (simply multiply each element of the matrix by the scalar quantity.) b a 11 a 12 a 21 a 22 = ba 11 ba 12 ba 21 ba 22 Example: For A = 1 2 3 4 5 6 and b = 4 calculate bA . Matrix multiplication: If A = [ a ij ] is an m × n matrix and B = [ b ij ] is an n × p matrix, then the product AB is an m × p matrix AB = [ c ij ] where: c ij = n X k =1 a ik b kj = a i 1 b 1 j + a i 2 b 2 j + a i 3 b 3 j + · · · + a in b nj
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