Introduction to Mathematical Economics Lecture 6

  Undergraduate Program _{Faculty of Economics & Business}

Universitas Padjadjaran Rational Function: Application

  

Previously

• Cubic Function: Application _•

  

Today

^• Totally review topics:

• ^– application of a 3x3 linear equation system in

  

System of Linear Equations

• If each constraint is expressed as a linear equation the constraints form a system of

  

System and Solution

^• The solution to a system of linear equations is a set of values which

  

System and Solutions Graphically

• ^• The solution of a system of two linear equations in two variables is represented graphically by the

  Solutions graphically

Equation 2

  Solutions graphically

  Solutions graphically Equation 2

  

System and Solution

^• The graphical description apply to systems

of equations with 2 variables, regardless

  

Solution possibilities

• ^• A system of linear equations having a unique _•

  solution is a consistent system of equation

  

Exactly, Overconstrained, and

Underconstrained Systems of Eq

• ^•

  Exactly constrained systems of equation

  are systems which have an equal number

  

Exactly, Overconstrained, and

Underconstrained Systems of Eq

• ^• Abbreviated way for describing the number of equations and variables in the system: A system
• ^• Solve the following system of three equations in the three variables X, Y, Z! _{Eq 1 : 2X + 3Y + Z = 6}

  Eliminate Z from the equation

  From equation 4 and 5:

  Eq 4 : 5X + 5Y = 6

  

Applications

• ^• An aircraft manufacturer uses 3 machines (X,Y, and Z) to manufacture three different parts,

  

Applications

^• To produce one part A requires the use of 3 hours of X, 4 hours of Y, and 1 hour of Z

  

Applications

  

3X + 4Y + Z = 380

• ^– In this form, X, Y, and Z represent the hours

  

Applications

  

2X + 4Y + 2Z = 400

_• Finally, each part C requires 6 hours of X,

  

Applications

^• To determine how many hours per month

each machine should be operated for the

  4Y + Z = 380

• _• Eq.2: 2X +

  

Applications

• Eq.1: 3X +

  4Y + 2Z = 400

  

Applications

• Thus, the system of 2 equations and 2 unknowns with Y eliminated is as follows:

  

Matrix Application

• Econometrics _• Input – Output table

  Topic Outline

• Basic Elements of Matrix Algebra _• Elementary Matrix Arithmetic

  Definition of a Matrix

• A matrix is a rectangular array of numbers enclosed in parentheses

  

Size / dimension / order

of a matrix

• ^• We describe a matrix size by specifying the

  Example of a matrix Elements of a matrix

• ^• We denote the element in row i and column j of matrix A, a _i,j

  Elements of a matrix

• ^• We denote the element in row I and column j of matrix A, a _i,j

  Elements of a matrix

• ^• We denote the element in row i and column j of matrix A, a _i,j

  Summary

  

Further definitions

• Square Matrix: matrix A is said to be square if _p,q

  Example

  Example

  Example Matrix arithmethic

• _– ^•

  Matrix Equality _{We say that matrix A is equal to matrix B if A is the p,q r,s}

  same size as B, i.e. p=r and q=s

  Matrix arithmethic

• _– ^•

  Matrix Addition _{We may add matrix A to matrix B if A is the same size}

  as B

  Matrix arithmethic _•

  Addition and Subtraction

  Matrix arithmethic

• Matrix Addition Laws _• Consider three p × q matrices, A, B and C

  A scalar is a single number

  Matrix arithmethic

• Scalar Multiplication of Matrices _–

  Matrix arithmethic

  Matrix arithmethic

• Scalar Multiplication Laws _{– Consider scalars, k and m and matrices A p,q and B p,q}
• –

  Matrix multiplication ^• Inner Product

• ^– Let a be a row vector and b be a column vector

  

Matrix multiplication

• For example, suppose that an individual consumes n goods.

  

Matrix multiplication

^– Inner product = scalar

  

Matrix multiplication

^• Two matrices, A and B have a product _{p,q r,s} only if q= r

  

Matrix multiplication

^• In order to form the matrix product AB the two matrices must be conformable

  C = A B

  (m x p) = (m x n) (n x p) (2 x 2) = (2 x 3) (3 x 2)

  7 1   1 2 3 

  



 C AB 8 2

    



  Matrix multiplication

  Matrix multiplication

  Matrix multiplication

  Matrix multiplication Matrix multiplication

• – _{outer product = matrix •}

  Inner & outer products

  Matrix multiplication

• Matrix Multiplication Laws _{– Consider matrices A p,q , B q, r , C q, r and D r, s}

  Matrix Arithmethic, in general