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

2. 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)

3. Catch
Catch: matrix algebra is only applicable to linear equation systems.
However, some transformation can be done to obtain a linear relation.
y = axb
log y = log a + b log x

4. Matrices and Vectors Example of a system of linear equations:
c1P1 + c2P2 = -c0
1P1 + 2P2 = -0
In general,
a11 x1 + a12 x2 +…+ a1nXn = d1
a21 x1 + a22 x2 +…+ a2nXn = d2
………………………………
am1 x1 + am2 x2 +…+ amnXn = dm
coefficients aij
variables x1, …,xn
constants d1, …,dm

5. Matrices as Arrays

6. Example:
6x1 + 3x2+ x3 = 22
x1 + 4x2+-2x3 =12
4x x2 + 5x3 = 10

7. 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=[aij]

8. 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 aij are ordered.
Special case: m = n, a square matrix

9. 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.

10. Matrix Notation
Ax = d
Questions: How do we multiply A and x? What is the meaning of equality?

11. Example Qd = Qs Qd = a - bP Q s= -c + dP can be rewritten as
1Qd + bP = a
Qs +-dP = -c

12. In matrix form…
Constant vector
Coefficient matrix
Variable vector

13. Matrix Operations
Addition and Subtraction: matrices must have the same dimensions
Example 1:
Example 2:

14. Matrix addition and subtraction
In general
Note that the sum matrix must have the same dimension as the component matrices.

15. Matrix subtraction
Subtraction
Example

16. Scalar Multiplication
To multiply a matrix by a number – by a scalar – is to multiply every element of that matrix by the given scalar.
Note that the rationale for the name scalar is that it scales up or down the matrix by a certain multiple. It can also be a negative number.

17. Matrix Multiplication
Given 2 matrices A and B, we want to find the product AB. The conformability condition for multiplication is that the column dimension of A (the lead matrix) must be equal to the row dimension of B ( the lag matrix).
BA is not defined since the conformability condition for multiplication is not satisfied.

18. Matrix Multiplication
In general, if A is of dimension m x n and B is of dimension p x q, the matrix product AB will be defined only if n = p.
If defined the product matrix AB will have the dimension m x q, the same number of rows as the lead matrix A and the same number of columns as the lag matrix B.

19. Matrix Multiplication
Exact Procedure

20. Matrix multiplication
Example : 2x2, 2x2, 2x2

21. Matrix multiplication
Example: 3x2, 2x1, 3x1

22. Matrix multiplication
Example: 3x3, 3x3, 3x3
Note, the last matrix is a square matrix with 1s in its principal diagonal and 0s everywhere else, is known as identity matrix

23. Matrix multiplication
from 4.4, p56
The product on the right is a column vector

24. Matrix multiplication
When we write Ax= d, we have

25. Simple national income model
Example : Simple national income model with two endogenous variables, Y and C
Y = C + Io + Go
C = a + bY
can be rearranged into the standard format
Y – C = Io – Go
-bY + C = a

26. Simple national income model
Coefficient matrix, vector of variables, vector of constants
To express it in terms of Ax=d,

27. Simple national income model
Thus, the matrix notation Ax=d would give us
The equation Ax=d precisely represents the original equation system.

28. Digression on  notation:
Subcripted symbols helps in designating the locations of parameters and variables but also lends itself to a flexible shorthand for denoting sums of terms, such as those which arise during the process of matrix multiplication.
j: summation index
xj: summand

29. Digression on  notation:

30. Digression on  notation:
The application of  notation can be readily extended to cases in which the x term is prefixed with a coefficient or in which each term in the sum is raised to some integer power.
general polynomial function

31. Digression on  notation:
Applying to each element of the product matrix C=AB

32. Digression on  notation:
Extending to an m x n matrix, A=[aik] and an
n x p matrix B=[bkj], we may now write the elements of the m x p matrix AB=C=[cij] as
or more generally,