1. 4.1 Matrix Operations What you should learn: Goal1 Goal2 Add and subtract matrices, multiply a matrix by a scalar, and solve the matrix equations. Use Matrices to solve real-life problems. 4.1 Matrix Operations

2. Vocabulary A matrix is a rectangular arrangement of numbers in rows and columns where the numbers are called entries. The dimensions of a matrix are given as the number of rows x the number of columns. Scalar multiplication is the process of multiplying each entry in a matrix by a scalar, a real number. 4.1 Matrix Operations

3. Adding and Subtracting Matrices + Solution To add or subtract matrices, they must have the same dimensions. Since is a 3 x 1 matrix and is a 1 x 3 matrix, you cannot add them. 4.1 Matrix Operations

4. Solution = = 4.1 Matrix Operations Adding and Subtracting Matrices

5. = + Cannot be done = = 4.1 Matrix Operations

6. Solving a Matrix Equation Solve the matrix equation for x and y: Multiply by -2x -10x = 20 x = -2 -16x = y -16(-2) = y 32 = y 4.1 Matrix Operations

7. 3x = 9 x = 3 2 = -y y = -2 Solve for x and y. 4.1 Matrix Operations

8. Solve for x and y. 2y + 5 = -5 2y = -10 y = -5 -6 + x = -7 x = -1 4.1 Matrix Operations

9. Using Matrix Operations Write a matrix that shows the average costs in health care from this year to next year. Comprehensive HMO Standard HMO Plus This Year (A) Next Year (B) Comprehensive HMO Standard HMO Plus Individual Family 4.1 Matrix Operations

10. Begin by adding matrix A and B to determine the total costs for two years. + = 4.1 Matrix Operations

11. Multiply the result by ½, which is equivalent to dividing by 2. Round your answers to the nearest cent to find the average. = 4.1 Matrix Operations

12. Using the matrix B on health care costs, write a matrix C for the following year that shows the costs after a 2% decrease. Multiply the matrix by.98 (1-.02) to get your reduction. = 4.1 Matrix Operations

13. Write a matrix which will show the monthly payment following a 3% increase in the costs from matrix B. Multiply the matrix by 1.03 to get your increase. = 4.1 Matrix Operations

14. Reflection on the Section What does it mean for a matrix to be a 4 x 3 matrix? assignment 4.1 Matrix Operations

15. 4.2 Multiplying Matrices What you should learn: Goal1 Goal2 Multiply two matrices Use Matrix Multiplication to solve real-life problems, such as finding the number of calories burned. 4.2 Multiplying Matrices

16. Goal1 Multiply two matrices 4.2 Multiplying Matrices When multiplying two matrices A times B, the number of columns in A must equal the number of rows B. 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.

17. Finding the Product of Two Matrices Because the number of columns in A equals the number of rows in B, the product is defined. AB will be a 3 x 2 matrix. 4.2 Multiplying Matrices

18. Multiply corresponding entries in the first row of A and the first column of B. Then add. Use similar procedure to write the other entries.

19. does not equal BA is undefined because B is a 2 x 2 matrix and A is a 3 x 2 matrix. The number of columns in B does not equal the number of rows in A. Y E S ! O r d e r i s i m p o r t a n t ! ! ! 4.2 Multiplying Matrices

20. Find the product. If it is not defined, state the reason. = = 4.2 Multiplying Matrices

21. Find the product. If it is not defined, state the reason. = = 4.2 Multiplying Matrices

22. Using Matrix Operations Multiply B by C first!!! Then multiply A by the result! 4.2 Multiplying Matrices

24. Use the given matrices to simplify the expression. AA A(B+C) AB + BC 4.2 Multiplying Matrices

25. Reflection on the Section If A is a 3x4 matrix and B is a 2x3 matrix, which product, AB or BA, is defined? Explain. assignment 4.2 Multiplying Matrices

26. 4.3 Determinants and Cramer’s Rule What you should learn: Goal1 Goal2 Evaluate determinants of 2x2 and 3x3 matrices. Use Cramer’s Rule to solve systems of linear equations. 4.3 Determinants and Cramer’s Rule

27. The determinant of a square matrix A is denoted by det A or |A|. Cramer’s Rule is a method of solving a system of linear equations using the determinant of the coefficient matrix of the linear system. The entries in the coefficient matrix are the coefficients of the variables in the same order. 4.3 Determinants and Cramer’s Rule

28. Reflection on the Section How do you find the determinant of a 2x2 matrix? assignment 4.3 Determinants and Cramer’s Rule

29. 4.4 Identity and Inverse Matrices What you should learn: Goal1 Goal2 Find and us inverse matrices. Use Inverse Matrices to solve real-life situations. 4.4 Identity and Inverse Matrices

30. Reflection on the Section How are a square matrix, its identity matrix, and the inverse matrix related? assignment 4.4 Identity and Inverse Matrices

31. 4.5 Solving Systems Using Inverse Matrices What you should learn: Goal1 Goal2 Solve systems of linear equations using inverse matrices Use systems of linear equations to solve real-life problems. 4.5 Solving Systems Using Inverse Matrices

32. Reflection on the Section How can you use inverse matrices to solve a system of equations? assignment 4.5 Solving Systems Using Inverse Matrices