Transcript Day 2
14.2 Matrix Multiplication OBJ: To find the product of two matrices MULTIPLYING TWO MATRICES A B AB 4 X 3 3 X 5 4X5 4 rows 3 rows 3 columns 5 columns MULTIPLYING TWO MATRICES A B AB 4 X 3 3 X 5 4X5 4 rows 4 rows 5 columns 5 columns MULTIPLYING TWO MATRICES If A is a 4 X 3 matrix and B is a 3 X 5 matrix, then the product AB is a 4 X 5 matrix. Matrices A and B can be multiplied only if the number of columns of A equal the number of rows of B. EX: Find the matrix product -3 2 1 2 0 0 4 3 -5 2 1 -1 _____________________________ _____________________________ [ ] EX: Let A = [ 2 -1 and B = 5 -1 0 -4 -2 7 3 ] • Is BA defined? Explain._____________ ______________________ ______________________ Show that (AB)t = BtAt BtAt = AB = [ 2 -1 3 ] 5 -1 0 -4 -2 7 _____________________ _____________________ _____________________ _____________________ _____________________ _____________________ _____________________ _____________________ AB = [ BtAt (AB)t = ] = Finding the Product of Two Matrices Find AB if –2 3 A = 1 –4 6 0 and B= –1 –2 SOLUTION Because A is a 3 X 2 matrix and B is a 2 X 2 matrix, the product AB is defined and is a 3 X 2 matrix. To write the entry in the first row and first column of AB, multiply corresponding entries in the first row of A and the first column of B. Then add. Use a similar procedure to write the other entries of the product. 3 4 Finding the Product of Two Matrices A B AB 3X2 2X2 3X2 –2 3 1 –4 6 0 –1 3 –2 4 (– 2)(– 1) + (3)(– 2) (– 2)(3) + (3)(4) (1)(–1) + (– 4)(–2) (1)(3) + (– 4)(4) (6)(– 1) + (0)(– 2) (6)(3) + (0)(4) Finding the Product of Two Matrices A B AB 3X2 2X2 3X2 –2 3 1 –4 6 0 –1 3 –2 4 (– 2)(– 1) + (3)(– 2) (– 2)(3) + (3)(4) (1)(–1) + (– 4)(–2) (1)(3) + (– 4)(4) (6)(– 1) + (0)(– 2) (6)(3) + (0)(4) Finding the Product of Two Matrices A B AB 3X2 2X2 3X2 –2 3 1 –4 6 0 –1 3 –2 4 (– 2)(– 1) + (3)(– 2) (– 2)(3) + (3)(4) (1)(–1) + (– 4)(–2) (1)(3) + (– 4)(4) (6)(– 1) + (0)(– 2) (6)(3) + (0)(4) Finding the Product of Two Matrices A B AB 3X2 2X2 3X2 (– 2)(– 1) + (3)(– 2) (– 2)(3) + (3)(4) (1)(– 1) + (– 4)(– 2) (1)(3) + (– 4)(4) (6)(– 1) + (0)(– 2) (6)(3) + (0)(4) –4 6 7 – 13 –6 18 EX: Machine I and Machine II produce items X, Y, Z at the hourly rate given in matrix H. Matrix D gives the number of hours each machine runs during the week. H= I II D= M T W Th F X3 2 I 8 8 8 7 7 Y5 4 II 6 10 12 11 9 Z1 2 Notice that in Example 2 the product H 3x2 . D2x5 is a 3 x 5 matrix. However, the product D2x5 . H3x2 is not defined because there are more elements in each row of D than there are in each column of H _ a. Give the dimensions of H, D, and HD. ________________________ b. Find HD. What information does HD give? M T W Th F [ ] ________________________ ________________________ c. How many Y items are produced on Monday? How many Z items are produced on Thursday? _________________________ _________________________ _________________________ _________________________ EX: The juniors at Adams High School held a two day bake sale to raise money for a class trip. Cupcakes, cookies, and pies were sold for $0.40, $0.25, $4.00 respectively. The first day 84 cupcakes, 210 cookies, and 27 pies were sold. The second day, 95 cupcakes, 184 cookies, and 17 pies were sold. Display this information in matrix form. Then use matrix multiplication to find the amount of money raised each day and altogether. Using Matrices to Calculate the Total Sales CLASS TRIP The juniors at Adam’s High School held a two-day bake sale to raise money for a class trip. 1st Day 84 cupcakes 210 cookies 27 pies 2nd Day 95 cupcakes 184 cookies 17 pies Each cupcake costs $0.40, each cookie costs $0.25, and each pie costs $4.00. Use matrix multiplication to find the total amount of money raised each day and altogether. Using Matrices to Calculate the Total Sales SOLUTION Write the bake sale items and costs per item in matrix form. Use matrix multiplication to find the total sales. Set up matrices so that columns of the bake sale items matrix match rows of the cost matrix. BAKE SALE ITEMS Cupcakes Cookies 1st Day 2nd Day 84 95 210 184 COST Dollars Pies 27 17 Cupcakes Cookies Pies 0.40 0.25 4.00 Using Matrices to Calculate the Total Sales Total money raised each day can be obtained by multiplying the bake sale items matrix by the cost per item matrix. The bake sale item matrix is 2 X 3 and the cost per item matrix is 3 X 1. Their product is a 2 X 1 matrix. 84 210 27 95 184 17 .40 .25 = 4.00 84(.40) + 210(.25) + 27(4) = 95(.40) + 184(.25) + 17(4) 194.1 152 Using Matrices to Calculate the Total Sales TOTAL Sales Dollars The labels of the product are: 194.1 Second Day 152 First Day The junior class raised 346.10 for their class trip.