**Example 1.** Consider
the matrix ,

**1
(a)** Find .

**Solution 1 (a).**

We want to find

First look at some powers

Now use the calculation

Find the expression for the general term

Find matrix exponential will
be the sum of the infinite series

The sum of the first few terms are:

Each element in can
be calculated by the sum of an infinite series and *Mathematica*
can assist us in these computations.

Therefore, the matrix exponential is

This can be compared to the matrix
exponential that
can be computed by using *Mathematica*'s built in
procedure **MatrixExp[At]**.

**Caveat.** This shows
the power of "artificial intelligence" that is available in
*Mathematica*. For this example the matrix had a full
set of eigenvectors. *Mathematica* is "smart" enough
to know how to compute the matrix exponential for the difficult case
when the set of eigenvectors is deficient.

(c) John H. Mathews 2004