1. Operations with Matrices: Multiplication Honors Math – Grade 8

2. Get ready for the Lesson The matrix below shows the monthly payments of six homeowners who took home-improvement loans for a period of 2 years at an annual interest rate of 6%. What matrix can be used to show the total amount owed by each homeowner? To find the matrix that shows the total amount owed by each homeowner for 2 years, multiply the given matrix by 24, the number of months in the loan. Multiplying a matrix by a real number is called scalar multiplication. To multiply a matrix by a real number, multiply each element by that number.

3. This matrix represents the total amount owed by each homeowner.

4. Multiply the matrix by the real number. To multiply a matrix by a real number, multiply each element by that number. In this example, we must multiply first before subtracting the matrices.

5. Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. Find AB KEY CONCEPT Multiplying Two Matrices To multiply two matrices… 1.Multiply the elements of each row of the first matrix by the corresponding elements in each column of the second matrix. 2.Find the sum of each set of products; each sum takes the corresponding position in the product matrix. The product of an m x n matrix and an n x r matrix is an m x r matrix. Let’s determine the dimensions of the product matrix first. AB

6. Find AB Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. The first matrix has two columns and the second matrix has two rows. These matrices can be multiplied. Multiply the elements of each row of the first matrix by the corresponding elements in each column of the second matrix. Add the products to find the product matrix.

7. Find the product matrix if it exists. If it does not exist, tell why not. Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. Determine if the product matrix exists by checking the dimensions. The product matrix does exist. Multiply the elements of each row of the first matrix by the corresponding elements in each column of the second matrix. Add the products to find the product matrix.

8. Find the product matrix if it exists. If it does not exist, tell why not. Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. Determine if the product matrix exists by checking the dimensions. The product matrix does exist. Multiply the elements of each row of the first matrix by the corresponding elements in each column of the second matrix. Add the products to find the product matrix.

9. Find the product matrix if it exists. If it does not exist, tell why not. Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. Determine if the product matrix exists by checking the dimensions. The product matrix does not exist because there are 3 columns in the first matrix and two rows in the second matrix. The columns of the first matrix must match the rows of the second matrix.

10. Mia and Dana make jewelry to sell at boutique shops. This year, they have crafted bracelets made from onyx, coral and silver beads. Their cost for each bead was %0.75, for onyx, $1.10 for coral and $0.50 for silver. Mia used 125 onyx, 100 coral, and 150 silver beads. Dana used 175 onyx, 115 coral, and 200 silver beads. What was the total cost of the beads they used to make these bracelets? Setup matrix A to show how many beads of each type each partner used and matrix B to show the cost per bead for each type. Find the product matrix AB to determine each partner’s cost for beads used. Add the elements of AB to find the total cost of the beads used.