Advanced Algebra Chapter 4

1
Advanced Algebra Chapter 4

• Matrices and Determinants ?

2
Matrix Operations4.1

• Matrix
• Rectangular array of numbers
• Dimensions
• Rows x Columns
• Entries
• Elements

3
Comparing Matrices

• Two matrices are equal iff their dimensions are
  equal and all corresponding entries are equal

4
Adding and Subtracting Matrices

• Must have the same dimensions to be compatible

5
Adding and Subtracting Matrices

• Example

6
Adding and Subtracting Matrices

• Example

7
Adding and Subtracting Matrices

• Example

8
Scalar Multiplication

• Any matrix can be multiplied by a real number
  constant, called a scalar
• This is called scalar multiplication

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Scalar Multiplication

• Example

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Scalar Multiplication

• Example

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Matrix Equations
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Matrix Equations
13
Properties of Matrix Operations

• Let A, B, and C be matrices with the same
  dimension and let c be a scalar.
• Associative Property of Addition
• (A B) C A (B C)

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Properties of Matrix Operations

• Let A, B, and C be matrices with the same
  dimension and let c be a scalar.
• Commutative Property of Addition
• A B B A

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Properties of Matrix Operations

• Let A, B, and C be matrices with the same
  dimension and let c be a scalar.
• Distributive Property of Addition
• c(A B) cA cB

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Properties of Matrix Operations

• Let A, B, and C be matrices with the same
  dimension and let c be a scalar.
• Distributive Property of Subtraction c(A B)
  cA - cB

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p.203 15-23, 33-36
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Multiplying Matrices4.2
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Multiplying Matrices

• The product of two matrices A and B is defined
  provided the number of columns in A is equal to
  the number of rows in B.
• Inner dimensions must be equal
• Row 1 times column 1 gives entry 1,1
• Row 2 times column 1 gives entry 2,1
• Etc.

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Multiplying Matrices
21
Multiplying Matrices
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Multiplying Matrices
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Multiplying Matrices
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Properties of Matrix Mult.

• Associative Property
• A(BC) (AB)C
• Left Distributive
• A(B C) AB BC
• Right Distributive
• (A B)C AC BC
• Associative Property of Scalar Multiplication
• c(AB) (cA)B A(cB)

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p.211 23-29
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Determinants4.3
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Determinants

• Numerical answer
• The only time working with matrices that you will
  get an answer that is not a matrix
• Can only find determinants of square matrices
• Used for many matrix applications
• Solving equations
• Determining if inverses exist
• And many, many more! ?

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Determinant of a 2 x 2

• The determinant of a 2 x 2 matrix is the
  difference of the products of the entries on the
  diagonals

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Determinants

• Example

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Determinants

• Example

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Determinant of a 3 x 3

• Repeat the 1st two columns to the right of the
  matrix, then subtract the product of the postive
  diagonals from the negative diagonals

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Determinants

• Example

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Determinants

• Example

34
Determinants

• Example

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p.21817-24 All
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Cramers Rule4.3 (Day 2)
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Cramers Rule

• Coefficient Matrix
• A matrix consisting of only the coefficients of a
  linear system

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Cramers Rule for 2 x 2 matrices

• Let A be the coefficient matrix of the linear
  systemIf det A does not equal 0, then the
  system has exactly 1 solution. The solution is

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Cramers Rule

• Example

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Cramers Rule

• Example

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Cramers Rule

• Example

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Cramers Rule for 3 x 3 matrices

• Let A be the coefficient matrix of the linear
  systemIf det A does not equal 0, then the
  system has exactly 1 solution. The solution is

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Cramers Rule

• Example

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Cramers Rule

• Example

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p.219 39-41, 45-47
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Extra Credit After Chp 4 Quiz

• 5 bales of hay are weighed two at a time in all
  possible combininations. The weights in pounds
  are 110, 111, 112, 113, 115, 116, 117, 118, 120,
  and 121. How much does each bale weigh?

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Identity and Inverse Matrices4.4
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Identity Matrix

• Identity
• Matrix with all 1s on the main diagonal and all
  0s in the remaining places
• The product of any matrix and the identity matrix
  is the original matrix

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Identity Matrix

• Example

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Identity Matrix

• Example

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Inverse Matrices

• Two matrices are inverses of each other iff their
  product produces the identity matrix
• Same idea as

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Inverse Matrices

• The inverse of the matrix
  is
• When will an inverse not exist?
• When det(A) 0

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Inverse Matrices

• Example

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Inverse Matrices

• Example

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Matrix Equations

• Example

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Matrix Equations

• Example

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p.227 13-16, 25-27, 33-36
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Using Codes4.4 Day 2
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Cryptograms

• Cryptogram
• Message written in code
• Military use
• Computer programming
• Internet

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Converting Messages

• Need to assign every letter a number
• Encode the message using a matrix
• Decode the matrix using the inverse of that matrix

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Coding

• H E L L O _
• Encoder matrix

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Uncoding

• Take resulting matrices and multiply by the
  inverse of the coding matrix to find original
  letters

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Assignment

• With a partner, you must create a message and
  encrypt it for another group. All groups will
  use the coding matrix

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Solving Systems Using Matrices4.5
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Solving Systems

• Set up an equation similar to AX B as we did in
  the previous section
• -Matrix of Variables
• -Matrix of Constants

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Solving Systems

• To solve the systemWe have to set this up in
  terms of a coefficient matrix, variable matrix,
  and an answer matrix

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Solving Systems

• Next, we multiply both sides by the inverse of
  the coefficient matrix

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Solving Systems

• Simplify

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Solving Systems

• Example

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Solving Systems

• Example

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p.233 23-31
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Augmented Matrices

• Augmented Matrix
• Matrix containing the coefficient matrix and the
  matrix of constants

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How do we use augmented matrices?

• Treat these like systems of equations
• Interchange two rows
• Multiply a row by a nonzero constant
• Add a multiple of 1 row to another

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What are we looking for?

• Get our matrix so we only have 1s on the main
  diagonal and only 0s underneath them

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Augmented Matrices

• Example
• System
• Matrix

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Augmented Matrices

• Example

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Augmented Matrices

• Example

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p.238 1-8