40S Applied Math Mr. Knight – Killarney School Slide 1 Unit: Matrices Lesson: MAT-3 Matrix Multiplication Matrix Operations Learning Outcome B-4 MAT-L3 Objectives: To perform Matrix Multiplication.

40S Applied Math Mr. Knight – Killarney School Slide 2 Unit: Matrices Lesson: MAT-3 Matrix Multiplication If Ted were to stop at a fast food store to buy a burger, two small fries, and three small soft drinks, and the prices were $2.50, $1.00, and $1.25 respectively, his total bill could be calculated as follows: Total Cost = 1* * *1.25 = $8.25 This calculation could be done using matrices. Write the order (the number of burgers, fries, and drinks) as a row matrix O: and the prices as column matrix B: Then the product of the two matrices represents the total cost of the order: Theory – Matrix Multiplication

40S Applied Math Mr. Knight – Killarney School Slide 3 Unit: Matrices Lesson: MAT-3 Matrix Multiplication Theory – Matrix Multiplication Note how each element of the first matrix is multiplied by each element of the second matrix. This process could be extended to include more than one order. If three people placed orders for burgers, fries, and drinks as shown on the matrix A: and the prices remained the same The total cost of each order could be found by multiplying matrices A and B in the following manner:

40S Applied Math Mr. Knight – Killarney School Slide 4 Unit: Matrices Lesson: MAT-3 Matrix Multiplication Therefore, Ted pays $8.25, Sam pays $6.25, and Bec pays $11.00 Note the way in which the calculations were completed. The elements of the first row of the first matrix were multiplied by the elements in the first column of the second matrix and added. This process is repeated for all the rows of the first matrix and the column of the second matrix. Theory – Matrix Multiplication

40S Applied Math Mr. Knight – Killarney School Slide 5 Unit: Matrices Lesson: MAT-3 Matrix Multiplication A toy manufacturing company produces three types of wooden toys: race cars, sailboats, and tractors. Matrix A shows the number of hours of labour for each type of toy. The R, S, and T represent the Race cars, Sailboats and Tractors, respectively. The following orders were received for the months of January and February: The shop manager of the manufacturing company wants to know how many hours of labour (cutting, assembling, and painting) were spent during the months of January and February. We can find this answer by multiplying matrix A and matrix B. Theory – Matrix Multiplication

40S Applied Math Mr. Knight – Killarney School Slide 6 Unit: Matrices Lesson: MAT-3 Matrix Multiplication Matrix C shows the number of hours spent on each type of operation. For example, 1390 hours were spent on painting in January. Theory – Matrix Multiplication

40S Applied Math Mr. Knight – Killarney School Slide 7 Unit: Matrices Lesson: MAT-3 Matrix Multiplication Note the following pattern: The column headings of the first matrix are the same as the row headings of the second matrix. When you multiply the above matrices, the column headings of the first matrix cancel the row headings of the second matrix. The row heading of the answer matrix is the same as the row heading of the first matrix. The column headings of the answer matrix are the same as the column headings of the second matrix. As you can see by the above example, there is quite a bit of computational work involved when multiplying matrices, and so we will use IT to perform the operations in the future. Theory – Matrix Multiplication

40S Applied Math Mr. Knight – Killarney School Slide 8 Unit: Matrices Lesson: MAT-3 Matrix Multiplication Note: When two matrices are multiplied: Matrix multiplication is "row by column" multiplication, where the elements of each row of the first matrix are multiplied by the elements of each column of the second matrix and added, as shown below: The number of columns in the first matrix must be the same as the number of rows in the second matrix. For matrices A ij and B mn, A x B works only if 'j' = 'm'. The product of two matrices is a matrix that has the same number of rows as the first matrix and the same number of columns as the second matrix. For example, if C is the product matrix for A x B, then A ij x B mn = C in. When solving problems using matrix multiplication, the units (for example, race cars, sailboats, and tractors) of the elements in the columns of the first matrix must be the same as the units of the elements in the rows of the second matrix. Theory – Matrix Multiplication

40S Applied Math Mr. Knight – Killarney School Slide 9 Unit: Matrices Lesson: MAT-3 Matrix Multiplication We can find the sums of the columns of any matrix A by multiplying it by a row matrix where each element is 1 and the dimensions are 1 x n where 'n' is the number of rows of matrix A. The Cosmetic Sales Company has three salespeople. Matrix A shows the gross sales for February for each salesperson. We can determine the sales total for each week by multiplying matrix A by a 1 x 3 row matrix B where each element is the number 1. and then determine BA, which we will label matrix C Therefore, the sales for Week 1 are $ , for Week 2 are $ , and so on. Special Case I – Summing Columns

40S Applied Math Mr. Knight – Killarney School Slide 10 Unit: Matrices Lesson: MAT-3 Matrix Multiplication We can find the sums of the rows of any matrix A by multiplying it by a column matrix where each element is 1 and the dimensions are n x 1 where 'n' is the number of columns of matrix A. The Cosmetic Sales Company has three salespeople. Matrix A shows the weekly incomes for the three salespeople. We can now determine the total sales for each person by multiplying matrix A by a 1 x 4 column matrix where each element is 1. First, we create column matrix D. and then determine AD, which we will name matrix E. Therefore, the February sales for Sara are $ , for Barb are $ , and Jeff are $ Special Case II – Summing Rows

40S Applied Math Mr. Knight – Killarney School Slide 11 Unit: Matrices Lesson: MAT-3 Matrix Multiplication We can adjust the values of all the elements in any column of matrix A by first creating a square matrix S that has the same dimensions as the number of columns in matrix A, and writing the elements in the main diagonal showing the adjustments. All the elements not in the main diagonal of matrix S are zeros. The Cosmetic Sales Company has different sales promotions each week, and so the commission rates for the salespeople are different each week. The rates for February are shown in the table below: Week 1Week 2Week 3Week 4 45%55%35%50% We want to determine the weekly income of each person. We can do this by first creating a 4 x 4 square matrix where the elements in the main diagonal are the weekly commission rates written in decimal form, and all the other elements are zeros. We will label this matrix F. We now multiply A and F to determine the weekly incomes. We will label this matrix G. Therefore, Sara earns $ during the first week, Jeff earns $ during the third week, and so on. Special Case III – Adjusting Values in each Column

40S Applied Math Mr. Knight – Killarney School Slide 12 Unit: Matrices Lesson: MAT-3 Matrix Multiplication A contractor develops a site by building nine 2-storey houses and six bungalows. On average, one 2-storey house requires 1600 units of materials and 2000 hours of labour, and one bungalow requires 1500 units of materials and 1800 hours of labour. Each unit of material costs $30.00, and labour costs $18.00 per hour. Use matrix operations to answer the following questions. Test Your Knowledge Hint: Work Backwards. Determine the dimensions of your final matrix, then derive the dimensions of the matrices being multiplied.