SECTION 11.4 Matrix Algebra 799 Some of the algebraic properties of scalar multiplication are listed next. 3 Find the Product of Two Matrices Unlike the straightforward definition for adding two matrices, the definition for multiplying two matrices is not what might be expected. In preparation for the definition, we need the following definitions: Note that a row vector and a column vector can be multiplied if and only if they both contain the same number of entries. NOTE R is a 1 by 3 matrix and C is a 3 by 1 matrix. j The Product of a Row Vector and a Column Vector If [ ] = − R 3 5 2 and = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ C 3 4 5 , then [ ] ( ) ( ) = − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ =⋅+− +−=− − =− RC 3 5 2 3 4 5 3 3 5425 92010 21 EXAMPLE 6 Properties of Scalar Multiplication Suppose h and k are real numbers, and A and B are m by n matrices. Then ( ) ( ) ( ) ( ) = + = + + = + k hA kh A k h A kA hA k A B kA kB • • • DEFINITION Product of a Row Vector and a Column Vector A row vector R is a 1 by n matrix [ ] = R r r rn 1 2 A column vector C is an n by 1 matrix = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ C c c cn 1 2 The product RC of R times C is defined as the number [ ] = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ = + + + RC r r r c c c r c r c r c n n n n 1 2 1 2 1 1 2 2
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