# Complexity Explorer Santa Fe Institute ## Vector and Matrix Algebra

• Introduction to this Tutorial
• Sets and Set Notation
• Proof of the Infinitude of Primes
• Boolean Set Operations
• Vectors and Vector Operations
• Matrices
• Matrix Operations
• Essential Types of Matrices
• Vector Spaces
• Determinants
• Eigenvalues and Eigenvectors
• Diagonalization and Powers
• Geometric Transformations
• Differentiation as a Matrix Operation
• Markov Chains

#### 3.4 Markov Chains » Quiz #21

Suppose that there are three states with the following migration patterns. Those who live in state A, remain there with probability .7, while with probability .2 they will move to state B and with probability .1 they will move to state C. Of those who live in state B, 10% move to state A, 60% stay in state B and 30% move to state C. And finally, of those who live in state C, 15% move to state A, 65% Initially the population of the states (respectively) are in millions: 10, 20, 300. In the long run, what will the population be of these three states?

Find a stochastic (i.e. transition matrix for this problem); find the long-term behabior of the population of the three states by solving for the steady-state vector.   (See next segment for solution.)