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4-3 Matrix Multiplication

4-3 Matrix Multiplication. Objectives: To multiply by a scalar To multiply two matrices. Objectives. Multiplying a Matrix by a Scalar Multiplying Matrices. Vocabulary. You can multiply a matrix by a real number. The real number factor (such as 3) is called a scalar. Vocabulary.

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4-3 Matrix Multiplication

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

  7. Properties of Scalar Multiplication

  8. 6 9 –12 15 27 –18 30 6 –3Y + 2 = 12 18 –24 30 27 –18 30 6 –3Y + = Scalar multiplication. 27 –18 30 6 12 18 –24 30 12 18 –24 30 –3Y = – Subtract from each side. 15 –36 54 –24 –3Y = Simplify. Multiply each side by – and simplify. 15 –36 54 –24 –5 12 –18 8 1 3 Y = – = 1 3 Solving Matrix Equations with Scalars 6 9 –12 15 27 –18 30 6 Solve the equation –3Y + 2 = .

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

  10. –2 5 3 –1 4 –4 2 6 = (–2)(4) + (5)(2) = 2 –2 5 3 –1 4 –4 2 6 2 = (–2)(4) + (5)(6) = 38 –2 5 3 –1 4 –4 2 6 2 38 = (3)(4) + (–1)(2) = 10 Multiplying Matrices –2 5 3 –1 4 –4 2 6 Find the product of and . Multiply a11 and b11. Then multiply a12 and b21. Add the products. The result is the element in the first row and first column. Repeat with the rest of the rows and columns.

  11. –2 5 3 –1 4 –4 2 6 2 38 10 = (3)(–4) + (–1)(6) = –18 –2 5 3 –1 4 –4 2 6 2 38 10 –18 The product of and is . Continued (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. 2 rows 4 rows 3 columns 2 columns Equal Dimensions of product matrix 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. QP (3 4) (3 3) not equal Determining Whether a Product Matrix Exists 3 –1 2 5 9 0 0 1 8 6 5 7 0 2 0 3 1 1 –1 5 2 Use matrices P = and Q = . Determine whether products PQ and QP are defined or undefined. Find the dimensions of each product matrix. PQ (33) (34) 34 product equal matrix 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.

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

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