Matrix Operations McDougal Littell Algebra 2 Larson, Boswell, Kanold, Stiff Larson, Boswell, Kanold, Stiff Algebra 2: Applications, Equations, Graphs Algebra 2: Applications, Equations, Graphs

Review:What is a Matrix? Definition of Matrix: A rectangular arrangement of numbers in rows and columns. Ex): Matrix A below has two rows and three columns. A= [ ] 2 rows [ ] 3 columns Note: * The DIMENSIONS of matrix A are 2 X 3 (read “2 by 3” ) * The numbers in a matrix are its ENTRIES. Ex.) The entry in the second row and third column is 5.

Multiplying Matrices If A is an m X n matrix and B is an n X p matrix, then the product AB is an m X p matrix. A * B = AB m X n n X p m X p Conclusion: The product of two matrices is defined iff the number of columns in A is equal to the number of rows in B.

Let’s Practice: Product defined? State whether the product AB is defined. Explain. Ex.) A: 3X4, B:4X5 Solution: Yes, the product is defined by definition of the product of two matrices. Ex.) A: 3X2, B:5X2 Solution: No, the product is not defined since the number of columns of A does not equal the number of rows of B.

Properties of Matrix Multiplication Let A, B, and C be matrices and let c be a scalar. 1.) Associative Property of Matrix Multiplication: A(BC) = (AB) C 2.) Left Distributive Property: A(B + C) = AB + AC 3.) Right Distributive Property: (A + B)C = AC + BC 4.) Associative Property of Scalar Multiplication: c(AB) = A(cB)

Finding the Product of Two Matrices Ex.) Find AB if A = [ -2 3 ] and B = [ -1 3] [ 1 -4 ] [ -2 4] [ 6 0 ] Solution: AB = [(-2)(-1) + (3)(-2) (-2)(3) + (3)(4)] [(1)(-1) + (-4)(-2) (1)(3) + (0)(4) ] [(6)(-1) + (0)(-2) (6)(3) + (0)(4)] = [ -4 6] [ 7 -13] [-6 18]

Finding the Product of Two Matrices If A = [3 2] and B = [1 -4], find each product. [-1 0] [2 1] Ex.) AB Solution: AB = [3 2][1 -4] = [7 -10] [-1 0][2 1] [-1 4] Ex.) BA Solution: BA = [1 -4][3 2] = [7 2] [2 1][-1 0] = [5 4] NOTE: AB DOES NOT EQUAL TO BA

Guided Practice Problems: Note to the teacher: See section 4.2: Multiplying Matrices on the McDougal Littell Algebra II book. Choose the appropriate guided-practice problems for your students.

Solutions to the Guided Practice Problems: Note to the teacher: Include the solutions to the guided-practice problems.