1 / 15

Ch. 4 Linear Models & Matrix Algebra

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.

latif
Download Presentation

Ch. 4 Linear Models & Matrix Algebra

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 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.

  2. 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,

  3. 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;

  4. Solving for X x=A-1d , where A-1 is the inverse (matrix) of A

  5. Inverse A-1of Matrix of A • Inverse of A is A-1 • AA-1 = A-1A = I • We are interested in A-1 becausex=A-1d

  6. Derivation of matrix inverse formula A-1 = adjoint A/|A|, where |A| = ai1ci1 + …. + aincin (Determinant) And, adjoint A = transposedcofactor matrix of A

  7. Determinant, Cofactor, and Minor

  8. How to get Determinant of A? By Laplace Expansion of cofactors, and minors in case the first row is used.

  9. Pattern of the signs for cofactor minors

  10. Adjoint of A: the transposed cofactor matrix

  11. Calculating Adjoint is hard! Is there any easier way to solve for x or specifically one of x, that is, xi ?

  12. 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!

  13. Solving for x1 using Cramer’s 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

  14. Solving for x1 / d1

  15. What about Comparative Statics?

More Related