Slide 7- 1

Chapter 7 Systems and Matrices

7.1 Solving Systems of Two Equations

Slide 7- 4 Quick Review

Slide 7- 5 What you’ll learn about The Method of Substitution Solving Systems Graphically The Method of Elimination Applications … and why Many applications in business and science can be modeled using systems of equations.

Slide 7- 6 Solution of a System A solution of a system of two equations in two variables is an ordered pair of real numbers that is a solution of each equation.

Slide 7- 7 Example Using the Substitution Method

Slide 7- 8 Example Using the Substitution Method

Slide 7- 9 Example Solving a Nonlinear System Algebraically

Slide Example Using the Elimination Method

Slide Example Finding No Solution

Slide Example Finding Infinitely Many Solutions

7.2 Matrix Algebra

Slide Quick Review

Slide What you’ll learn about Matrices Matrix Addition and Subtraction Matrix Multiplication Identity and Inverse Matrices Determinant of a Square Matrix Applications … and why Matrix algebra provides a powerful technique to manipulate large data sets and solve the related problems that are modeled by the matrices.

Slide Matrix

Slide Matrix Vocabulary Each element, or entry, a ij, of the matrix uses double subscript notation. The row subscript is the first subscript i, and the column subscript is j. The element a ij is the ith row and the jth column. In general, the order of an m × n matrix is m×n.

Slide Example Determining the Order of a Matrix

Slide Matrix Addition and Matrix Subtraction

Adding Matrices

Slide Example Matrix Addition

Slide Example Using Scalar Multiplication

Slide The Zero Matrix

Slide Additive Inverse

Slide Matrix Multiplication

Let A denote an m by r matrix and let B denote an r by n matrix. The product AB is defined as the m by n matrix whose entry in row i, column j is the product of the ith row of A and the jth column of B. Multiplying Matrices Note: If we multiply a matrix by a constant, this is equivalent to multiplying each term in the matrix by the constant.

Slide Example Matrix Multiplication

Slide Example Matrix Multiplication

Example: Find the product AB

The Identity Matrix

Slide Identity Matrix

Slide Inverse of a Square Matrix

Slide Inverse of a 2 × 2 Matrix

Slide Determinant of a Square Matrix

Slide Inverses of n × n Matrices An n × n matrix A has an inverse if and only if det A ≠ 0.

Slide Example Finding Inverse Matrices

Slide Properties of Matrices Let A, B, and C be matrices whose orders are such that the following sums, differences, and products are defined. 1. Community property Addition: A + B = B + A Multiplication: Does not hold in general 2. Associative property Addition: (A + B) + C = A + (B + C) Multiplication: (AB)C = A(BC) 3. Identity property Addition: A + 0 = A Multiplication: A·I n = I n ·A = A 4. Inverse property Addition: A + (-A) = 0 Multiplication: AA -1 = A -1 A = I n |A|≠0 5. Distributive property Multiplication over addition: A(B + C) = AB + AC (A + B)C = AC + BC Multiplication over subtraction: A(B - C) = AB - AC (A - B)C = AC - BC

Slide Matrices and Transformations P(x,y)  P’(x’,y’) 

Slide Matrices and Transformations Rotation through an angle  The rotation through an angle  maps each point P(x,y) in the rectangular coordinate plane to the point P’(x’,y’). where

Slide Matrices and Transformations or

Slide Matrices and Transformations Find the rotation matrix about the origin whose angle is  /3.

Slide Matrices and Transformations Where does the point (4,-2) move?

7.3 Multivariate Linear Systems and Row Operations

Slide Quick Review

Slide What you’ll learn about Triangular Forms for Linear Systems Gaussian Elimination Elementary Row Operations and Row Echelon Form Reduced Row Echelon Form Solving Systems with Inverse Matrices Applications … and why Many applications in business and science are modeled by systems of linear equations in three or more variables.

Slide Equivalent Systems of Linear Equations The following operations produce an equivalent system of linear equations. 1. Interchange any two equations of the system. 2. Multiply (or divide) one of the equations by any nonzero real number. 3. Add a multiple of one equation to any other equation in the system.

Slide Row Echelon Form of a Matrix A matrix is in row echelon form if the following conditions are satisfied. 1. Rows consisting entirely of 0’s (if there are any) occur at the bottom of the matrix. 2. The first entry in any row with nonzero entries is The column subscript of the leading 1 entries increases as the row subscript increases.

Slide Elementary Row Operations on a Matrix A combination of the following operations will transform a matrix to row echelon form. 1. Interchange any two rows. 2. Multiply all elements of a row by a nonzero real number. 3. Add a multiple of one row to any other row.

Slide Example Finding a Row Echelon Form

Slide Example Finding a Row Echelon Form

Slide Reduced Row Echelon Form If we continue to apply elementary row operations to a row echelon form of a matrix, we can obtain a matrix in which every column that has a leading 1 has 0’s elsewhere. This is the reduced echelon form.

Slide Example Solving a System Using Inverse Matrices

Slide Example Solving a System Using Inverse Matrices

Slide Multivariate Linear Systems and Row Operations Page 602

Slide Multivariate Linear Systems and Row Operations Page 602

Slide Multivariate Linear Systems and Row Operations

7.4 Partial Fractions

Slide Quick Review

Slide What you’ll learn about Partial Fraction Decomposition Denominators with Linear Factors Denominators with Irreducible Quadratic Factors Applications … and why Partial fraction decompositions are used in calculus in integration and can be used to guide the sketch of the graph of a rational function.

Slide Partial Fraction Decomposition of f(x)/d(x)

Slide Example Decomposing a Fraction with Distinct Linear Factors

Slide Example Decomposing a Fraction with Distinct Linear Factors

Slide Example Decomposing a Fraction with an Irreducible Quadratic Factor

Slide Example Decomposing a Fraction with an Irreducible Quadratic Factor

7.5 Systems of Inequalities in Two Variables

Slide Quick Review Solutions

Slide What you’ll learn about Graph of an Inequality Systems of Inequalities Linear Programming … and why Linear programming is used in business and industry to maximize profits, minimize costs, and to help management make decisions.

Slide Steps for Drawing the Graph of an Inequality in Two Variables 1. Draw the graph of the equation obtained by replacing the inequality sign by an equal sign. Use a dashed line if the inequality is. Use a solid line if the inequality is ≤ or ≥. 2. Check a point in each of the two regions of the plane determined by the graph of the equation. If the point satisfies the inequality, then shade the region containing the point.

Slide Example Graphing a Linear Inequality

Slide Example Graphing a Linear Inequality

Slide Example Solving a System of Inequalities Graphically

Slide Example Solving a System of Inequalities Graphically

Slide Chapter Test

Slide Chapter Test

Slide Chapter Test

Slide Chapter Test

Slide Chapter Test Solutions

Slide Chapter Test Solutions

Slide Chapter Test Solutions

Slide Chapter Test Solutions