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

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) a+b=(-2,7) x=(2,-1,5), c = 7 (a constant) cx =(14,-7,35)

5. Matrices Recall linear equations, where the goal is to solve for x,y, and z: 2x+y+z=1 5x-y+7z=0 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.

9. Special Matrices If the number of rows equals the number of columns (m=n), then the matrix is said to be square A matrix containing all zeros is called a zero matrix A matrix with 0’s everywhere but with 1’s on the diagonal is an identity matrix

11. Matrix Multiplication

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-1=I If the inverse exists, there is only one

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 <- solve(A)

18. Representing a system of equations 3x-2y+z=1 -x+7y-4z=-5

19. Markov Matrices