954 CHAPTER 9 Systems and Matrices The proofs of the following properties of scalar multiplication are left for Exercises 95–98. Scalar Multiplication In work with matrices, a real number is called a scalar to distinguish it from a matrix. The product of a scalar k and a matrix X is the matrix kX, each of whose elements is k times the corresponding element of X. S Now Try Exercises 41 and 43. EXAMPLE 4 Multiplying Matrices by Scalars Find each product. (a) 5c 2 0 -3 4d (b) 3 4 c 20 12 36 -16d SOLUTION (a) 5c 2 0 -3 4d = c 5122 5102 51-32 5142 d Multiply each element of the matrix by the scalar 5. = c 10 0 -15 20d (b) 3 4 c 20 12 36 -16d = c 3 41202 3 41122 3 41362 3 41-162d Multiply each element of the matrix by the scalar 3 4. = c 15 9 27 -12d These screens support the results in Example 4. Properties of Scalar Multiplication Let A and B be matrices of the same dimension, and let c and d be scalars. Then these properties hold true. 1 c +d2A =cA +dA 1cA2d = 1cd2A c1A +B2 =cA +cB 1cd2A =c1dA2 Matrix Multiplication We have seen how to multiply a real number (scalar) and a matrix. The product of two matrices can also be found. To illustrate, we multiply A = c -3 5 4 0 2 4d and B = £ -6 2 3 4 3 -2§ . First locate row 1 of A and column 1 of B, which are shown shaded below. A = c -3 5 4 0 2 4d B = £ -6 2 3 4 3 -2§ Multiply corresponding elements, and find the sum of the products. -31-62 + 4122 + 2132 = 32 This result is the element for row 1, column 1 of the product matrix. 3 4 2
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