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Unformatted text preview: 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 , A = ( a 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 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....
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