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 Vectors Matrices as Arrays Vectors 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 -1 d, where A-1 is the inverse (matrix) of A

5 Inverse A -1 of Matrix of A Inverse of A is A -1 AA -1 = A -1 A = I We are interested in A -1 because x=A -1 d

6 Derivation of matrix inverse formula A -1 = adjoint A / |A|, where |A| = a i1 c i1 + …. + a in c in (Determinant) And, adjoint A = transposed cofactor 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, x i ?

12 Cramer's Rule for each of x, say, x 1 : “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 x 1 using Cramer’s rule Find the determinant |A| Find the determinant |A 1 | where d i is the constant vector substituted for the 1 st col. X 1 = |A 1 |/|A| Repeat for X 2 by substituting the constant vector for the 2 nd col. And solving for |A 2 | and so on as necessary

14 Solving for x 1 / d 1

15 What about Comparative Statics?