Im having trouble doing questions 1 and 4 on the following sheet, would someone please help get started, thanks!
1. consider the column vector $\displaystyle {\bold{p}}=[p_t, p_b]'$ the first element of which is the probability that the student travelled by train yesterday, and the second the probability that they travelled by bus yesterday. Then from the statement of the problem we know that:
$\displaystyle
{\bold{A}}\left[ \begin{array}{c} 1 \\ 0 \end{array} \right] = \left[ \begin{array}{c} 1/3 \\ 2/3 \end{array} \right]
$
So the first column of $\displaystyle {\bold{A}}$ is $\displaystyle [1/3, 2/3]'$.
Similarly by considering its action on $\displaystyle [0,1]'$ we can conclude that the second column of $\displaystyle {\bold{A}}$ is $\displaystyle [2/5, 3/5]'$.
So:
$\displaystyle
{\bold{A}} = \left[ \begin{array}{cc} 1/3 & 2/5 \\ 2/3 & 3/5 \end{array} \right]
$
That's you started.
RonL