Review of Some Basic Concepts from Matrix Algebra econ 222 lecture notes

1
Review of Some Basic Concepts from Matrix
Algebra econ 222 lecture notes 1

• Petra Todd

2
Vectors

• A number can be used to represent a point on the
  real line
• A pair of numbers can be used to represent a
  point in two-dimensional space
• Consider n-space, for example, earnings in n
  different years

3
Examples

• a(1,2), b(-3,5)
• ab(-2,7)
• x(2,-1,5), c 7 (a constant)
• cx (14,-7,35)

4
(No Transcript)
5
Matrices

• Recall linear equations, where the goal is to
  solve for x,y, and z
• 2xyz1
• 5x-y7z0
• Will write in terms of matrices

6
Definition of a matrix
7
Rows, Columns, Elements of a Matrix

• The rows and columns of a matrix are vectors
• Rows are (1,1,-2), (-1,4,-5)
• Columns are (1,-1),(1,4),(-2,-5)
• We call the i,j cell of a matrix aij and refer to
  the ij element of component.

8
(No Transcript)
9
Special Matrices

• If the number of rows equals the number of
  columns (mn), then the matrix is said to be
  square
• A matrix containing all zeros is called a zero
  matrix
• A matrix with 0s everywhere but with 1s on the
  diagonal is an identity matrix

10
(No Transcript)
11
Matrix Multiplication
12
(No Transcript)
13
(No Transcript)
14
Properties of Matrix Multiplication
15
Inverse of a matrix

• If A is a square matrix, then can define the
  inverse, denoted by A-1
• An inverse for A is a matrix AA-1I
• If the inverse exists, there is only one

16
(No Transcript)
17

• An inverse can be tedious to compute, especially
  for large matrices
• Luckily, computers can do the work for you.
• In R, the command to find an inverse is
• Ainv lt- solve(A)

18
Representing a system of equations

• 3x-2yz1
• -x7y-4z-5

19
Markov Matrices
20
(No Transcript)
21
(No Transcript)