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Notes-Chapter 4-Part 1

# Notes-Chapter 4-Part 1 - ALGEBRA Chapter4...

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LINEAR MODELS AND MATRIX  ALGEBRA Chapter 4 Alpha Chiang, Fundamental Methods  of Mathematical Economics 3 rd  edition

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Why Matrix Algebra As more and more commodities are included  in models, solution formulas become  cumbersome. Matrix algebra enables to do us many things:   provides a compact way of writing an equation  system leads to a way of testing the existence of a  solution by evaluation of a determinant gives a method of finding solution (if it exists)
Catch Catch: matrix algebra is only applicable  to linear equation systems.   However, some transformation can be  done to obtain a linear relation. y = ax b log y = log a + b log x

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Matrices and Vectors Example of a system of linear equations: c 1 P 1 + c 2 P 2 = -c 0 γ 1 P 1 + γ 2 P 2 = - γ 0 In general, a 11 x 1 + a 12 x 2 +…+ a 1n X n = d 1 a 21 x 1 + a 22 x 2 +…+ a 2n X n = d 2 ……………………………… a m1 x 1 + a m2 x 2 +…+ a mn X n = d m coefficients a ij variables x 1, …, x n constants d 1 , …,d m
Matrices as Arrays    11 12 1 1 1 21 22 2 2 2 1 2 n n m m mn n a a a x d a a a x d A x d a a a x dm = = = L L L L L L M M L

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Example: 6x 1 + 3x 2 + x 3 = 22 x 1 + 4x 2 +-2x 3 =12 4x 1 - x 2 + 5x 3 = 10 1 2 3 6 3 1 22 1 4 2 12 4 1 5 10 x A x x d x = - = = -
Definition of Matrix A matrix is defined as a rectangular array of numbers, parameters, or variables. Members of the array are termed elements of the matrix. Coefficient matrix: A=[a ij ] 1,2,..., 1,2,..., i m j n = =

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Matrix Dimensions Dimension of a matrix = number of rows x number of columns, m x n m rows n columns Note: row number always precedes the column number. this is in line with way the two subscripts are in a ij are ordered. Special case: m = n , a square matrix
Vectors as Special Matrices one column : column vector one row:row vector usually distinguished from a column vector by the use of a primed symbol: Note that a vector is merely an ordered n- tuple and as such it may be interpreted as a point in an n-dimensional space.

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Matrix Notation Ax = d Questions: How do we multiply A and x ?
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Notes-Chapter 4-Part 1 - ALGEBRA Chapter4...

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