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**Unformatted text preview: **Math 006 (Lecture 8) Addition, Subtraction and Scalar Multiplication
Deﬁnition 1. Two matrices are equal if they have the same size and their corresponding elements are equal. Deﬁnition 2. A matrix with elements that are all zeros is called a zero matrix. Deﬁnition 3. The addition/subtraction of two matrices of the same size is the matrix with elements that are the addition/subtraction of the corresponding elements of the two given matrices. Deﬁnition 4. The product of a number k with a matrix M , denoted by kM , is a matrix formed by multiplying each element of M by k . Example 1. Evaluate the following expressions: 2 −3 12 3 −1 (b) − 0 (a) (c) 3 12 −2 4 0 2 −2 3 −1 − 4 1 −1 3 2 −2 −1 7 5 0 −2 + 1 −1 1 −3 8 2 −2 + 0 −5 Matrix Product
• The product of a 1 × n row matrix and given by b1 b2 a1 a2 · · · an . . . bn an n × 1 column matrix is a 1 × 1 matrix = a1 b 1 + a2 b 2 + · · · an b n . • If A is an m × n matrix and B is a p × n matrix, the matrix product of A and B , denoted by AB , is an m × n matrix whose element in the ith row and j th column is the number obtained from the product of the ith row of A and j th column of B . • If the number of columns in A does not equal to the number of rows in B , the matrix product AB is not deﬁned. 1 Example 2. ab (a) cd 2 (b) 1 −1 Evaluate the following matrix products: AB CD 1 1 −1 0 1 0 2 1 20 0 (c) 1 −1 0 1 2 1 20 2 6 −1 −3 21 1 0 −1 0 12 36 (d) (e) 12 36 2 6 −1 −3 (f) 2 −3 0 −5 2 −2 −5 (g) 2 −2 2 −3 0 Applications
Example 3. A nutritionist for a cereal company blends two cereals in three diﬀerent mixes. The amounts of protein, carbohydrate and fat (in grams per ounce) in each cereal are given below. Cereal A Cereal B Mix X Mix Y Mix Z Protein 4g/oz 2g/oz Cereal A 15g/oz 10g/oz 5g/oz Carbohydrate 20g/oz 16g/oz Cereal B 5g/oz 10g/oz 15g/oz Fat 3g/oz 1g/oz (a) Find the amount of protein in Mix X. (b) Find the amount of fat in Mix Z. 2 ...

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