Ch. 4 Linear Models

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Ch. 4 Linear Models Matrix Algebra

• Matrix algebra can be used
• a.       To express the system of equations in a
  compact manner.
• b.      To find out whether solution to a system
  of equations exist.
• c.       To obtain the solution if it exists.

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4.1 Matrices and VectorsMatrices as
ArraysVectors as Special Matrices

• Matrix is a rectangle array of parameter,
  coefficients, etc.
• A general form matrix Ax d,

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Step 1 Write in matrix format
A x d
A parameter matrix x variable column
vector d constant column vector A general form
matrix Ax d, solve for x
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Solving for X x A-1 d , where A-1 is the
inverse (matrix) of A
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Inverse A-1of Matrix of A

• Inverse of A is A-1
• AA-1 A-1A I
• We are interested in A-1 because xA-1d

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Derivation of matrix inverse formula

• A-1 adjoint A / A,
• where
• A ai1ci1 . aincin (Determinant)
• And, adjoint A
• transposed cofactor matrix of A

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• Determinant, Cofactor, and Minor

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How to get Determinant of A?
By Laplace Expansion of cofactors, and minors in
case the first row is used.
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• Pattern of the signs for cofactor minors

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Adjoint of A the transposed cofactor matrix
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• Calculating Adjoint is hard!
• Is there any easier way to solve for x or
  specifically one of x, that is, xi ?

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Cramer's Rule for each of x, say, x1 The
easy way

• The numerator represents a determinant of A in
  which the ith column is replaced by the vector of
  constants, i.e., no need to invert A!

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Solving for x1 using Cramers rule

• Find the determinant A
• Find the determinant A1 where di is the
  constant vector substituted for the 1st col.
• X1 A1/A
• Repeat for X2 by substituting the constant vector
  for the 2nd col. And solving for A2 and so on
  as necessary

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Solving for x1 / d1
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What about Comparative Statics?