In order to raise a matrix to some power, use the MatrixPower command.  Here is the syntax, along with a couple of examples.

?MatrixPower

a = (({{1, 2, 1}, {-1, 2, 0}, {0, 4, 2}}));
MatrixForm[a]

MatrixForm[a.a]
MatrixForm[MatrixPower[a,2]]
MatrixForm[a.a.a]
MatrixForm[MatrixPower[a,3]]
MatrixForm[MatrixPower[a,20]]

Here is the transformation matrix “trans” that we used when discussing the Markov chain for the coin flip game (discussed on p. 301 of the textbook).

trans = (({{1, .5, 0, 0, 0, 0}, {0, 0, .5, 0, 0, 0}, {0, .5, 0, .5, 0, 0}, {0, 0, .5, 0, .5, 0}, {0, 0, 0, .5, 0, 0}, {0, 0, 0, 0, .5, 1}}));
MatrixForm[trans]

MatrixForm[
MatrixPower[trans, 5] ]

We can use the Manipulate command to easily see how things change as some paramter changes. Below is the documentation, plus the example of the transformation matrix raised to some power.

?Manipulate

Manipulate[MatrixForm[
MatrixPower[trans, exponent] ], {exponent, 1, 50, 1}]

It seems the best way to display numbers to a fixed number of decimal places is to use the NumberForm command, specifically the second example listed in the documentation below.  (Note that it behaves a bit unexpectedly when it comes to very small numbers and scientitfic notation, as you will see in examples.)

?NumberForm

NumberForm[MatrixForm[
MatrixPower[trans, 5] ], {10,3}]

This shows our transformation matrix raised to a power, and we have modified the number format and made it so that the exponent also appears to the right of the slider bar.

Manipulate[NumberForm[MatrixForm[
MatrixPower[trans, exponent] ], {4,3}], {exponent, 1, 60, 1,Appearance -> "Labeled"}]