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Chiang_Ch5

Chiang_Ch5 - Ch 5 Linear Models& Matrix Algebra Ch 5...

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Unformatted text preview: Ch. 5 Linear Models & Matrix Algebra Ch. 5 Linear Models & Matrix Algebra 5.1 Conditions for Nonsingularity of a Matrix 5.2 Test of Nonsingularity by Use of Determinant 5.3 Basic Properties of Determinants 5.4 Finding the Inverse Matrix 5.5 Cramer's Rule 5.6 Application to Market and National-Income Models 5.7 Leontief Input-Output Models 5.8 Limitations of Static Analysis 1 5.1 5.1 Conditions for Nonsingularity of a Matrix Conditions for Nonsingularity of a Matrix 3.4 3.4 Solution of a General-equilibrium System (p. 44) Solution of a General-equilibrium System (p. 44) x + y = 8 x + y = 9 (inconsistent & dependent) 2x + y = 12 4x + 2y= 24 (dependent) 2x + 3y = 58 y = 18 x + y = 20 (over identified & dependent) = 9 8 1 1 1 1 y x = 24 12 2 4 1 2 y x 2 = 20 18 58 1 1 1 3 2 y x 5.1 5.1 Conditions for Non-singularity of a Matrix Conditions for Non-singularity of a Matrix 3.4 3.4 Solution of a General-equilibrium System (p. 44) Solution of a General-equilibrium System (p. 44) Sometimes equations are not consistent, and they produce two parallel lines. (contradict) Sometimes one equation is a multiple of the other. (redundant) 3 y x x + y = 9 x + y = 8 y 12 For both the equations Slope is -1 5.1 Conditions for Non-singularity of a Matrix 5.1 Conditions for Non-singularity of a Matrix Necessary versus sufficient conditions Necessary versus sufficient conditions Conditions for non-singularity Conditions for non-singularity Rank of a matrix Rank of a matrix A) Square matrix , i.e., n. equations = n. unknowns. Then we may have unique solution. (nxn , necessary) B) Rows (cols.) linearly independent (rank=n, sufficient) A & B (nxn, rank=n) (necessary & sufficient), then nonsingular 4 5.1 Elementary Row Operations 5.1 Elementary Row Operations (p. (p. 86) 86) Interchange any two rows in a matrix Multiply or divide any row by a scalar k (k ≠ 0) Addition of k times any row to another row These operations will: ◦ transform a matrix into a reduced echelon matrix (or identity matrix if possible) ◦ not alter the rank of the matrix ◦ place all non-zero rows before the zero rows in which non-zero rows reveal the rank 5 5.1 5.1 Conditions for Nonsingularity of a Matrix Conditions for Nonsingularity of a Matrix Conditions for non-singularity, Rank of a matrix (p. 86) Conditions for non-singularity, Rank of a matrix (p. 86) 6 -- + --- --- +- --...
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Chiang_Ch5 - Ch 5 Linear Models& Matrix Algebra Ch 5...

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