# Matrix question involving probability

$\begin{pmatrix} 0.8 & 0.1 & 0.1 \\ 0.4 & 0.5 & 0.1 \\ 0.6 & 0.2 & 0.2 \end{pmatrix}$

Let A, B and C be three brands of icecream. $a_{ij}$ represents the probability of a person move from brand i to j. The main diagonal represents the chance of a person to stay in the same brand. What is the matrix that represents the probabilities after two questionares?

Answer: $$\begin{pmatrix} 0.74 & 0.15 & 0.11 \\ 0.58 & 0.31 & 0.11 \\ 0.68 & 0.20 & 0.12 \end{pmatrix}$$

My idea was to sum each $a_{ij}$ to itself but obviously this didn`t worked. Excuse my english.

## 2 Answers

Square the first matrix to get the second.

• Why multiply? If to change brand you just need to change one time so you just sum, right? – João Pedro Oct 20 '15 at 20:10

In case this is a question posed to you completely out of context, this is known as a Markov chain. In two jumps there are three paths to get from I to j. You have to sum the probabilities of each possible path.

• This exercise is proposed in linear algebra course. I too think that is out of context. So summing the paths leads to the result, are you sure? – João Pedro Oct 20 '15 at 20:23
• Summing the paths is the same as squaring the matrix. If squaring the matrix gives the correct answer, so will summing the paths. – Tim Galvin Oct 20 '15 at 20:40