How to compute future's (tomorrow's) distribution given today's? Marital status can be defined as single, married, separated, or divorced.
Today's distribution is:
Single = 49%
Married = 25%
Separated = 15%
Divorced = 11%

I am supposed to find the distribution for tomorrow. I am also given a table simply entitled "Table 1":
             Single'    Married'    Separated'    Divorced'
Single        0.85       0.12         0.02          0.01
Married       0          0.88         0.08          0.04
Separated     0          0.13         0.45          0.42
Divorced      0          0.09         0.02          0.89

So how does one find "distribution of people tomorrow"? 
Thank you 
 A: Look at the  $4\times 4$ matrix towards the end. That is the second table, stripped of the header words. So it is just a $4\times 4$  square array of numbers. This matrix is called the transition matrix. Let us call it $M$.
Now let $v_0$ be the row vector consisting of the numbers in your first table, written as a row.
Then the row vector $v_1$ that gives the probabilities "tomorrow," or more likely next year, is given by
$$v_1=v_0M.$$
If you know about multiplying a row vector by a matrix, the rest should be easy.
But some background in Linear Algebra would be useful for full understanding. And for later work you may need information about other topics in Linear Algebra to deal effectively with transition matrices.  
Remarks: $1.$ In much of Linear Algebra, there is a preference for column vectors. To write things in that style, let $M^T$ be the transpose of the matrix we get from the second table. Write the initial distribution as a column vector $v_0$. Then we gt $v_1=M^Tv_0$.  
$2.$ To find the distribution vector $v_2$ after two time period, just multiply by $M$ (or $M^T$) again.  
