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

2. 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= [ 6 2 -1 ] 2 rows [-2 0 5 ] 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.

3. Special Matrices (plural) Row Matrix: Only 1- row ex.) [ 3 -2 0 4 ] Column Matrix: Only 1-column ex.) [ 1 ] [ 3 ] Square matrix: The same number of rows and columns. ex.) [ 4 -1 ] [ 2 0 ] Zero matrix: Matrix whose entries are all zeros. ex.) [ 0 0 ] [ 0 0 ]

4. Are this two boxes of chocolates equal? ? =

5. Why are this two boxes of chocolates equal? ? = How many rows of chocolates does each box have? How many columns of chocolates does each box have? Look at every chocolate of each box, a.) Are there the same number of chocolates in each box? Are there the same number of rows and same number of columns? b.) Are the chocolates in the corresponding positions the same? Then, what can you conclude about the boxes? How does this relate to matrices?

6. Equal matrices EQUAL MATRICES: Two matrices are EQUAL if their DIMENSIONS are the SAME and the ENTRIES in CORRESPONDING POSITIONS are EQUAL. ex.) [ 5 0 ] equal to? [ 5 0 ] [-4/4 3/4] [ -1.75] Solution: Yes! Corresponding entries are equal and both matrices have the same dimensions. ex.) [-2 6] equal to? [-2 6 ] [ 0 -3] [ 3 0 ] Solution: No! Corresponding entries are not equal, but both matrices have the same dimensions.

7. Adding and Subtracting Matrices Rule: Add or Subtract matrices if they have the SAME DIMENSIONS. Steps: To add or subtract matrices, you simply add or subtract corresponding entries. ex.) [ 3 ] [ 1 ] [-4] + [ 0 ] [ 7 ] [ 3 ] Solution: [ 3 + 1] [ 4 ] [-4 +0] = [-4 ] [7 + 3] [10 ] ex.) [ 8 3 ] _ [ 2 -7] [ 4 0 ] [ 6 -1] Solution: [ 8 – 2 3 – (-7)] = [ 6 10 ] [ 4 - 6 0 – (-1)] [-2 1 ]

8. What is a scalar? Definition of a SCALAR: A real number. Steps: To multiply a matrix by a scalar, simply multiply each entry in the matrix by the scalar. (Process called scalar multiplication.) ex.) Perform the indicated operation. [ -2 0] 3 [ 1 4] = [ 0 3] Solution: [3(-2) 3(0)] [-6 0] [3(1) 3(4)] = [ 3 12] [3(0) 3(3)] [ 0 9]

9. Solving a Matrix Equation: Ex.) Solve the matrix equation for x and y: [ -2X -8] equal to [ 6 y ] [-10 -9] [-10 -9] Solution: Notice that if this matrices are equal, then the entries in the corresponding positions are also equal. Therefore, -2X = 6 solve for x X = -3 and - 8 = y Thus, the value of X is -3 and y is -8.

10. Properties of Matrix Operations Let A,B, and C be matrices with the same dimensions and let c be a scalar. When adding matrices, you can regroup them and change their order without affecting the result. 1.) Associative Property of Addition: (A + B) + C = A = (B + C) 2.) Commutative Property of Addition: A + B = B + A Multiplication of a sum or difference of matrices by a scalar obeys the distributive property. 3.) Distributive Property of Addition: c(A + B) = cA + cB 4.) Distributive Property of Subtraction: c(A – B ) = cA - cB

11. Guided Practice Problems: Note to the teacher: See section 4.1: Matrix operations on the McDougal Littell Algebra II book. Choose the appropriate guided-practice problems for your students.

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