1. 4-3 Matrix Multiplication Objectives: To multiply by a scalar To multiply two matrices

2. Objectives Multiplying a Matrix by a Scalar Multiplying Matrices

3. Vocabulary You can multiply a matrix by a real number. The real number factor (such as 3) is called a scalar.

4. Vocabulary Scalar multiplication multiplies a matrix A by a scalar c. To find the resulting matrix cA, multiply each element of A by c. ex.

5. Mulitplying a Matrix by a Scalar Multiply the matrix by a scalar of -5 Do Quick Check #1 on the bottom of Page 182

6. Find the sum of –3M + 7N for M = and N =. 2 –3 0 6 –5 –1 2 9 –3M + 7N = –3 + 7 2 –3 0 6 –5 –1 2 9 = + –6 9 0 –18 –35 –7 14 63 = –41 2 14 45 Using Scalar Products Do Quick Check #2a & 2b on Page 183

7. Properties of Scalar Multiplication If A is an m x n matrix, then cA is also an m x n matrix Closure Properties (cd)A = c(dA)Associative Property C(A+B) = cA + cB (c+d)A = cA + dA Distributive Property 1(A) = AMultiplicative Identity 0 (A) = 0 and c(0) = 0Multiplicative Property of Zero

8. Solve the equation –3Y + 2 =. 6 9 –12 15 27 –18 30 6 –3Y + 2 = 6 9 –12 15 27 –18 30 6 –3Y + = Scalar multiplication. 12 18 –24 30 27 –18 30 6 –3Y = – Subtract from each side. 27 –18 30 6 12 18 –24 30 12 18 –24 30 –3Y = Simplify. 15 –36 54 –24 Y = – = 15 –36 54 –24 1313 –5 12 –18 8 Multiply each side by – and simplify. 1313 Solving Matrix Equations with Scalars

9. (continued) –3Y + 2 = 6 9 –12 15 27 –18 30 6 –3 + 2 Substitute. 6 9 –12 15 27 –18 30 6 –5 12 –18 8 + Multiply. 12 18 –24 30 27 –18 30 6 15 –36 54 –24 = Simplify. 27 –18 30 6 27 –18 30 6 Check: Continued

10. Find the product of and. Multiply a 11 and b 11. Then multiply a 12 and b 21. Add the products. –2 5 3 –1 4 –4 2 6 –2 5 3 –1 4 –4 2 6 = (–2)(4) + (5)(2) = 2 The result is the element in the first row and first column. Repeat with the rest of the rows and columns. –2 5 3 –1 4 –4 2 6 = (–2)(4) + (5)(6) = 38 2 –2 5 3 –1 4 –4 2 6 = (3)(4) + (–1)(2) = 10 2 38 Multiplying Matrices

11. (continued) 2 38 10 –2 5 3 –1 4 –4 2 6 = (3)(–4) + (–1)(6) = –18The product of and is. –2 5 3 –1 4 –4 2 6 2 38 10 –18 Continued

12. Dimensions of a Product Matrix If matrix A is an m x n matrix and matrix B is an n x p matrix, then the product matrix AB is an m x p matrix. Dimensions of product matrix Equal 2 rows 2 columns 3 columns 4 rows The dimensions of a product matrix AB are 4 x 3

13. Dimensions of a Product Matrix What are the dimensions of the product matrix AB?

14. Use matrices P = and Q =. Determine whether products PQ and QP are defined or undefined. 3 –1 2 5 9 0 0 1 8 6 5 7 0 2 0 3 1 1 –1 5 2 Find the dimensions of each product matrix. (3  3) (3  4) 3  4 PQ product equal matrix (3  4) (3  3) QP not equal Product PQ is defined and is a 3  4 matrix. Product PQ is undefined, because the number of columns of Q is not equal to the number of rows in P. Determining Whether a Product Matrix Exists

15. Homework Pg 186 # 1,2,3,4,9,11,12,20,21,22,23