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M313Lec04 - Math 313 Lecture#4 1.3 Matrix Arithmetic Part...

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Math 313 Lecture #4 § 1.3: Matrix Arithmetic, Part II § 1.4: Matrix Algebra, Part I More about Matrix Multiplication. Let A = ( a ij ) be an m × n matrix and B = ( b ij ) an n × r matrix. The ( i, j ) entry of the product AB is a scalar product: n k =1 a ik b kj = a ( i, :) b j . That is, as the index k varies from 1 to n , the values a ik are in the i th row of A , and the values b kj are in the j th column of B . This means that the ( i, j ) entry of AB is the scalar product of the i th row of A with the j th column of B (as shown above). Now as the ( i, j ) entry of AB is of the form a ( i, :) b j , then the j th column of AB is the column vector Ab j = a (1 , :) b j a (2 , :) b j . . . a ( m, :) b j = a 11 a 12 . . . a n 1 a 21 a 22 . . . a 2 n . . . . . . . . . . . . a m 1 a m 2 . . . a mn b 1 j b 2 j . . . b nj . Thus the columns of the product of A with B are AB = ( Ab 1 , Ab 2 , . . . , Ab n ) . The Rules of Matrix Algebra. Let α and β be scalars, and A , B , and C matrices for which the indicated operations are defined. Then 1.
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