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Chapter 6

Chapter 6. Matrices and Linear Systems. Objectives. In this chapter, you will… Review properties of real numbers Use matrices to organize information Add, subtract, and multiply matrices Solve systems of linear equations with matrices

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Chapter 6

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  1. Chapter 6 Matrices and Linear Systems

  2. Objectives In this chapter, you will… • Review properties of real numbers • Use matrices to organize information • Add, subtract, and multiply matrices • Solve systems of linear equations with matrices • Graph inequalities on a coordinate plane and solve systems of inequalities • Write and graph inequalities that represent conditions that must be met simultaneously.

  3. Key Terms in Chapter 6 • Matrices - rectangular arrangements of numbers to organize information and solve problems • Commutative - two numbers can be multiplied in either order and get the same results • Counterexample - a single case where a property doesn't hold true • Transition Diagrams - a diagram that shows how something changes from one time to the next • Dimensions - give the numbers of the rows and columns (rows x columns) • Feasible Region - the set of points that is the solution to a system of inequalities containing two variables

  4. Key Terms cont. • Entry/Element - a number in the matrix • Scalar Multiplication - the multiplication of a matrix by a number • Scalar - the number the matrix is multiplied by • Matrix Addition - the addition of two matrices • Matrix Multiplication - the multiplication of two matrices • Identity Matrix - the square matrix that does not alter the entries of a square matrix [A] under multiplication • Linear Programming - the process of finding a feasible region and determining the point that gives the maximum or minimum value to a specific expression

  5. Key Terms cont. • Inverse Matrix - the matrix that will produce an identity matrix when multiplied by [A] • Augmented Matrix - a single matrix that contains columns for the coefficients of each variable and a final column for the constant terms • Row Reduction Method - transforms and augmented matrix into a solution matrix • Reduced Row-Echelon Form - a matrix where each row is reduced to a 1 and a solution and the rest of the matrix entries are 0's • Inequality - an algebraic statement of a situation where there is a range of possible values • Constraints - all of the limitations of a system of inequalities • Vertex - the corner of a region that satisfies an inequality

  6. Matrices Have A Variety of Uses • Matrices are used to organize many kinds of information • Matrices provide ways to organize data about things such as inventory or the coordinates of vertices of a polygon. • They can be used to solve systems of equations. • They are used to show how numbers in two different categories change over time. • They can be used to organize the vertices of a geometric figure on a coordinate plane. • Once the information is organized, matrices can then be added, subtracted, and multiplied to help solve the problem.

  7. Example

  8. Example 2

  9. Example 3

  10. Matrices Can be Added, Subtracted, Multiplied 58 130 35 203 105 [A] + [B] =

  11. Matrices in the Real World • In the real world, a matrix can be used to display any kind of information. • Example: If a matrix had a table like this: It would be turned into a matrix like this: 9 16 8 11 14 12

  12. Matrices in the Real World cont. • Matrices are also used in computer programming and animation. • Examples: • http://www.youtube.com/watch?v=7yW6UKpyeM8 • http://www.mathwarehouse.com/algebra/matrix/matix-movie-effect.php

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