Copyright © Cengage Learning. All rights reserved. 2 SYSTEMS OF LINEAR EQUATIONS AND MATRICES

Copyright © Cengage Learning. All rights reserved. 2.5 Multiplication of Matrices

3 Matrix Product

4 To define matrix multiplication, let’s consider the following problem. On a certain day, Al’s Service Station sold 1600 gallons of regular, 1000 gallons of regular plus, and 800 gallons of premium gasoline. If the price of gasoline on this day was $3.09 for regular, $3.29 for regular plus, and $3.45 for premium gasoline, find the total revenue realized by Al’s for that day.

5 Matrix Product The day’s sale of gasoline may be represented by the matrix A = [ ] Next, we let the unit selling price of regular, regular plus, and premium gasoline be the entries in the matrix Row matrix (1  3) Column matrix (3  1)

6 Matrix Product The first entry in matrix A gives the number of gallons of regular gasoline sold, and the first entry in matrix B gives the selling price for each gallon of regular gasoline, so their product (1600)(3.09) gives the revenue realized from the sale of regular gasoline for the day. A similar interpretation of the second and third entries in the two matrices suggests that we multiply the corresponding entries to obtain the respective revenues realized from the sale of regular, regular plus, and premium gasoline.

7 Matrix Product Finally, the total revenue realized by Al’s from the sale of gasoline is given by adding these products to obtain (1600)(3.09) + (1000)(3.29) + (800)(3.45) = 10,994 or $10,994.

8 Matrix Product This example suggests that if we have a row matrix of size 1  n, A = [a 1 a 2 a 3 … a n ] and a column matrix of size n  1,

9 Matrix Product Then we may define the matrix product of A and B, written AB, by (12)

10 Example 1 Let A = [1 – 2 3 5] and Then

11 Matrix Product Returning once again to the matrix product AB in Equation (12), observe that the number of columns of the row matrix A is equal to the number of rows of the column matrix B. Observe further that the product matrix AB has size 1  1 (a real number may be thought of as a 1  1 matrix). Schematically,

12 Matrix Product More generally, if A is a matrix of size m  n and B is a matrix of size n  p (the number of columns of A equals the numbers of rows of B), then the matrix product of A and B, AB, is defined and is a matrix of size m  p. Schematically,

13 Example 3 Let Compute AB. Solution: The size of matrix A is 2  3, and the size of matrix B is 3  3.

14 Example 3 – Solution The product AB is given by cont’d

15 Matrix Product In general, AB  BA for two square matrices A and B. However, the following laws are valid for matrix multiplication.

16 Matrix Product The square matrix of size n having 1s along the main diagonal and 0s elsewhere is called the identity matrix of size n.

17 Matrix Product The identity matrix has the properties that I n A = A for every n  r matrix A and BI n = B for every s  n matrix B. In particular, if A is a square matrix of size n, then I n A = AI n = A

18 Example 5 Let Then

19 Example 5 so I 3 A = AI 3 = A, confirming our result for this special case. cont’d

20 Practice p. 122 Concept Questions #1 & 2