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## Systems and Matrices

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Title: Systems and Matrices

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Chapter 7
• Systems and Matrices

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7.1
• Solving Systems of Two Equations

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Quick Review

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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.

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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.

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Example Using the Substitution Method

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Example Using the Substitution Method

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Example Solving a Nonlinear System Algebraically

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Example Using the Elimination Method

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Example Finding No Solution

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Example Finding Infinitely Many Solutions

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7.2
• Matrix Algebra

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Quick Review

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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.

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Matrix

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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.

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Example Determining the Order of a Matrix

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Matrix Addition and Matrix Subtraction

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Adding Matrices
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Example Matrix Addition

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Example Using Scalar Multiplication

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The Zero Matrix

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Additive Inverse

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Matrix Multiplication

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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.
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Example Matrix Multiplication

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Example Matrix Multiplication

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Example Find the product AB
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The Identity Matrix
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Identity Matrix

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Inverse of a Square Matrix

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Inverse of a 2 2 Matrix

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Determinant of a Square Matrix

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Inverses of n n Matrices
• An n n matrix A has an inverse if and only if
• det A ? 0.

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Example Finding Inverse Matrices

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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

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Matrices and Transformations
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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
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Matrices and Transformations
or
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Matrices and Transformations
Find the rotation matrix about the origin whose
angle is ?/3.
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Matrices and Transformations
Where does the point (4,-2) move?
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7.3
• Multivariate Linear Systems and Row Operations

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Quick Review

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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.

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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.

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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.

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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.

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Example Finding a Row Echelon Form

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Example Finding a Row Echelon Form

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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.

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Example Solving a System Using Inverse Matrices

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Example Solving a System Using Inverse Matrices

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Multivariate Linear Systems and Row Operations
Page 602
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Multivariate Linear Systems and Row Operations
Page 602
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Multivariate Linear Systems and Row Operations
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7.4
• Partial Fractions

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Quick Review

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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.

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

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Example Decomposing a Fraction with Distinct
Linear Factors

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Example Decomposing a Fraction with Distinct
Linear Factors

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Example Decomposing a Fraction with an
Irreducible Quadratic Factor

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Example Decomposing a Fraction with an
Irreducible Quadratic Factor

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7.5
• Systems of Inequalities in Two Variables

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Quick Review Solutions

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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.

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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 lt
orgt. 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.

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Example Graphing a Linear Inequality

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Example Graphing a Linear Inequality

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Example Solving a System of Inequalities
Graphically

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Example Solving a System of Inequalities
Graphically

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Chapter Test

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Chapter Test

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Chapter Test

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Chapter Test

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Chapter Test Solutions

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Chapter Test Solutions

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Chapter Test Solutions

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Chapter Test Solutions
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