# 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.