Chapter 7: Systems of Equations and Inequalities Matrices

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Chapter 7 Systems of Equations and Inequalities
Matrices

• 7.1 Systems of Equations
• 7.2 Solution of Linear Systems in Three Variables
• 7.3 Solution of Linear Systems by Row
  Transformations
• 7.4 Matrix Properties and Operations
• 7.5 Determinants and Cramers Rule
• 7.6 Solution of Linear Systems by Matrix Inverses
• 7.7 Systems of Inequalities and Linear
  Programming
• 7.8 Partial Fractions

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7.5 Determinants and Cramers Rule

• Subscript notation for the matrix A
• The row 1, column 1 element is a11 the row 2,
  column 3 element is a23 and, in general, the row
  i, column j element is aij.

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7.5 Determinants of 2 2 Matrices

• Associated with every square matrix is a real
  number called the determinant of A. In this text,
  we use det A.

The determinant of a 2 2 matrix A, is defined
as
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7.5 Determinants of 2 2 Matrices

• Example Find det A if
• Analytic Solution Graphing Calculator Solution

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7.5 Determinant of a 3 3 Matrix

The determinant of a 3 3 matrix A, is
defined as
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7.5 Determinant of a 3 3 Matrix

• A method for calculating 3 3 determinants is
  found by re-arranging and factoring this formula.
• Each of the quantities in parentheses represents
  the determinant of a 2 2 matrix that is part of
  the
• 3 3 matrix remaining when the row and column
  of the multiplier are eliminated.

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7.5 The Minor of an Element

• The determinant of each 3 3 matrix is called a
  minor of the associated element.
• The symbol Mij represents the minor when the ith
  row and jth column are eliminated.

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7.5 The Cofactor of an Element
Let Mij be the minor for element aij in an n n
matrix. The cofactor of aij, written Aij, is

• To find the determinant of a 3 3 or larger
  square matrix
• Choose any row or column,
• Multiply the minor of each element in that row or
  column by a 1 or 1, depending on whether the
  sum of i j is even or odd,
• Then, multiply each cofactor by its corresponding
  element in the matrix and find the sum of these
  products. This sum is the determinant of the
  matrix.

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7.5 Finding the Determinant

• Example Evaluate det ,
  expanding
• by the second column.
• Solution First find the minors of each element in
  the
• second column.

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7.5 Finding the Determinant

• Now, find the cofactor.
• The determinant is found by multiplying each
  cofactor by its
• corresponding element in the matrix and finding
  the sum of
• these products.

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7.5 Cramers Rule for 2 2 Systems

• Note Cramers rule does not apply if D 0.

For the system where, if possible,
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7.5 Applying Cramers Rule to a System with Two
Equations

• Example Use Cramers rule to solve the system.
• Analytic Solution By Cramers rule,

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7.5 Applying Cramers Rule to a System with Two
Equations

• The solution set is
• Graphing Calculator Solution
• Enter D, Dx, and Dy as matrices A, B, and C,
• respectively.

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7.5 Cramers Rule for a System with Three
Equations
For the system where