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Chapter 7

- Systems and Matrices

7.1

- Solving Systems of Two Equations

Quick Review

What youll 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.

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.

Example Using the Substitution Method

Example Using the Substitution Method

Example Solving a Nonlinear System Algebraically

Example Using the Elimination Method

Example Finding No Solution

Example Finding Infinitely Many Solutions

7.2

- Matrix Algebra

Quick Review

What youll 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.

Matrix

Matrix Vocabulary

- Each element, or entry, aij, of the matrix uses
- double subscript notation. The row subscript is
- the first subscript i, and the column subscript

is - j. The element aij is the ith row and the jth
- column. In general, the order of an m n
- matrix is mn.

Example Determining the Order of a Matrix

Matrix Addition and Matrix Subtraction

Adding Matrices

Example Matrix Addition

Example Using Scalar Multiplication

The Zero Matrix

Additive Inverse

Matrix Multiplication

Multiplying Matrices

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.

Note If we multiply a matrix by a constant,

this is equivalent to multiplying each term in

the matrix by the constant.

Example Matrix Multiplication

Example Matrix Multiplication

Example Find the product AB

The Identity Matrix

Identity Matrix

Inverse of a Square Matrix

Inverse of a 2 2 Matrix

Determinant of a Square Matrix

Inverses of n n Matrices

- An n n matrix A has an inverse if and only if
- det A ? 0.

Example Finding Inverse Matrices

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 AIn InA A
- 4. Inverse property
- Addition A (-A) 0
- Multiplication AA-1 A-1A In 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

Matrices and Transformations

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

Matrices and Transformations

or

Matrices and Transformations

Find the rotation matrix about the origin whose

angle is ?/3.

Matrices and Transformations

Where does the point (4,-2) move?

7.3

- Multivariate Linear Systems and Row Operations

Quick Review

What youll 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.

Equivalent Systems of Linear Equations

- The following operations produce an equivalent
- system of linear equations.
- Interchange any two equations of the system.
- Multiply (or divide) one of the equations by any

nonzero real number. - Add a multiple of one equation to any other

equation in the system.

Row Echelon Form of a Matrix

- A matrix is in row echelon form if the following
- conditions are satisfied.
- Rows consisting entirely of 0s (if there are

any) occur at the bottom of the matrix. - The first entry in any row with nonzero entries

is 1. - The column subscript of the leading 1 entries

increases as the row subscript increases.

Elementary Row Operations on a Matrix

- A combination of the following operations will
- transform a matrix to row echelon form.
- Interchange any two rows.
- Multiply all elements of a row by a nonzero real

number. - Add a multiple of one row to any other row.

Example Finding a Row Echelon Form

Example Finding a Row Echelon Form

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 0s elsewhere. This is the
- reduced echelon form.

Example Solving a System Using Inverse Matrices

Example Solving a System Using Inverse Matrices

Multivariate Linear Systems and Row Operations

Page 602

Multivariate Linear Systems and Row Operations

Page 602

Multivariate Linear Systems and Row Operations

7.4

- Partial Fractions

Quick Review

What youll 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.

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

Example Decomposing a Fraction with Distinct

Linear Factors

Example Decomposing a Fraction with Distinct

Linear Factors

Example Decomposing a Fraction with an

Irreducible Quadratic Factor

Example Decomposing a Fraction with an

Irreducible Quadratic Factor

7.5

- Systems of Inequalities in Two Variables

Quick Review Solutions

What youll 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.

Steps for Drawing the Graph of an Inequality in

Two Variables

- Draw the graph of the equation obtained by

replacing the inequality sign by an equal sign.

Use a dashed line if the inequality is lt

orgt. Use a solid line if the inequality is

or . - 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.

Example Graphing a Linear Inequality

Example Graphing a Linear Inequality

Example Solving a System of Inequalities

Graphically

Example Solving a System of Inequalities

Graphically

Chapter Test

Chapter Test

Chapter Test

Chapter Test

Chapter Test Solutions

Chapter Test Solutions

Chapter Test Solutions

Chapter Test Solutions