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571_1.3 - 1.3/1.5 MATRIX ARITHMETIC/ELEMENTARY MATRICES...

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1.3/1.5 MATRIX ARITHMETIC/ELEMENTARY MATRICES KIAM HEONG KWA p. 27 The scalar a ij of a matrix A = ( a ij ) = a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n . . . . . . . . . a m 1 a m 2 · · · a mn is called the ( i , j ) entry of A . p. 29 Two m × n matrices A = ( a ij ) and B = ( b ij ) are said to be equal if and only if their corresponding entries agree: A = B iff a ij = b ij i, j. p. 29 (Scalar multiplication) For any matrix A = ( a ij ) and any scalar α , αA = ( αa ij ) . pp. 29-30 (Addition) The sum of two m × n matrices A = ( a ij ) and B = ( b ij ) is the matrix A + B = ( a ij + b ij ) . The difference of A and B is the matrix A + ( - 1) B . The m × n matrix 0 whose entries are all zero is called the zero matrix and it acts as an additive identity: A + 0 = 0 + A = A. The matrix - A = ( - 1) A is the additive inverse of A : A + ( - A ) = ( - A ) + A = 0 . pp. 38-39 The transpose of an m × n matrix A = ( a ij ) is the n × m matrix A T = ( b ij ) whose ( i, j ) entry is given by b ij = a ji . In particular, the transpose of a row vector is a column vector. It follows that a row vector ~ a i is the i th row of A if and only if ~ a T i is the i th column of A T . Also, ( A T ) T = A. A square matrix A is said to be symmetric if and only if A T = A . Date : June 18, 2011. 1
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2 KIAM HEONG KWA p. 35 (Matrix multiplication) Let A = ( a ij ) be an m × n matrix and B = ( b ij ) be an n × r matrix. The product of A and B is the m × r matrix AB = C = ( c ij ) whose ( i, j ) entry is c ij = n
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