Chapter 2 Solving Linear Systems

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Chapter 2 Solving Linear Systems

• Matrix Definitions
• Matrix--- Rectangular array/ block of numbers.
• The size/order/dimension of a matrix
• (The numbers of ROWS) by(x) (the numbers of
  COLUMNS)

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• ELEMENTS individual numbers of matrix
• aij --- an element of ROW i and COLUMN j
• SQURE matrix
• The numbers of ROWS the numbers of COLUMNS
• IDENTITY matrix symbol---I
• TRANSPOSED matrix Rows and columns of a matrix
  are switched

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• Matrix Operations
• Addition
• Two same size matrices can be added.
• CABBA

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• Multiplication
• Multiplication of a Matrix by a Scalar
• AkA
• Example
• Multiplication of 2 Matrices
• Two Matrix can be multiplied if and only if---
• The NUMBER OF COLUMNS OF THE FIRST MATRIX
  The NUMBER OF ROWS OF THE SECOND MATRIX
• The Size of the resultant matrix ---
• the NUMBER OF ROWS OF THE FIRST MATRIX by the
  NUMBER OF COLUMNS OF THE SECOND MATRIX

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• Example
• First Matrix Second Matrix Multipication
  Size
• 
  Possible?
• A B
  AB
• (a)(2x2) (2x2) YES
  (2x2)
• (b)(3x3) (3x2) YES
  (3x2)
• (c)(3x3) (2x3) NO
• (d)(5x5) (5x1) YES
  (5x1)

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• Notice that
• AB exists and so does BA with BA being (2x2)
• AB exists, BA does not exist as a (3x2) cannot be
  multiplied into a (3x3)
• AB does not exist, Its possible that BA exists
• How to calculate the elements of CAB
• Example

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• A---mxn matrix Iidentity matrix
• I A A
• A I A

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• Matrix Inversion
• Only Square matrices have the inverse but not all
  square matrices have inverses.
• Scalar number
• The inverse of matrix A is denoted by A-1
• The size of A-1 is the same as A and
• A A-1 I A-1 A
• Any Matrix times its own inverse is just the
  appropriately sized identity matrix

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• Matrix Equality
• Two matrices are said to be equal if
• They are same size
• Corresponding elements in the two matrices are
  the same

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• Break-Even Model in Matrix Algebra terms
• Break-even model in linear equations
• 1 TR 0 TC 20q 0
• 0 TR 1 TC 25q 500
• 1 TR 1 TC 0q 0
• Let

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• Axb ? A-1 Ax A-1 b
• ? I x A-1 b
• ? x A-1 b
• Example

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• Modelling Steps
• Set up the system of linear equations
• Decide upon an order in which to express the
  unknowns
• The unknowns on the LHS of the equations
• Identify the following 3 matrices
• A Square matrix of coefficients relating to the
  unknowns
• x the matrix of unknows
• b the matrix of RHS constants
• Find matrix inverse A-1 of A
• Perform the matrix multiplication A-1b
• Use the matrix equality rule to find the elements
  of x
• Give the business interpretation of x

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