MG211.1: Solutions for Chapter 4
4.1 Setting up a new TV channel for TV station CCB involves a number of activities.
These are identified below, together with the cost per day for each activity (in 000) and
its the normal time to completion (in days). How
MG211.1: Solutions for Chapter 7
7.1. Revisit Exercise 1.4: What is the expected number of times you have to roll a die
to get a 6? Solve the problem by using Markov chains.
Solution. Let us first define the problem as a Markov chain. We have two states:
Preliminary Review Question
Suppose the RREF of a matrix A has the form
What can you say about the system of equations Ax = b?
definition as a cofactor expansion
If A = (C|d), What are the solutions of Cx =
Lecture 4 Elementary Matrices
When does A1 exist?
Elementary row operations:
RO1 multiply a row by a non-zero constant
Find A1 .
RO2 interchange two rows.
RO3 add a multiple of one row to another.
Suppose B is a 3 3 matrix, e.g.
1 0 1
At the end of week zero, distribution of T = 20, 000:
How to solve linear systems of dierence equations
10, 000 = 0.5T
What is a Markov Chain?
8, 000 = 0.4T
2, 000 = 0.1T
Number of shoppers at the end of the next week?
First, review the basics.
(a) Put the matrix A into reduced row echelon form.
Look through each weeks lecture notes and exercise set.
(b) If A represents the augmented matrix of a system of equations,
Bx = b, what are the equations
If |A| = 0, what does this tell you?
Ax = 0 = 0 x
has a non-trivial solution
What is orthogonal diagonalization?
So 0 is an eigenvalue of A
When can we do it?
How can we use it?
|A| = 0
v is an eigenvector corresponding to = 0
But rst . . .
What can go wrong?
= 2 + 1 = 0
|A I| =
Solve systems of linear dierential equations
y = Ay
A has eigenvalues which are not real.
Lecture 17 Applications