Discrepancy over matrix exponential

Question

I am trying to compute $\large e^A$ for $A = \left( \begin{array}{ccc} 0 & a \\ 0 & 0 \end{array} \right)$

Using $\large e^A = \sum \limits_{k=0}^\infty \frac{1}{k!} A^k$

Writing out the first few terms of this, I get...

$\left( \begin{array}{ccc} 1 & 0 \\ 0 & 1 \end{array} \right) + \left( \begin{array}{ccc} 0 & a \\ 0 & 0 \end{array} \right) + \left( \begin{array}{ccc} 0 & 0 \\ 0 & 0 \end{array} \right) + . . . $

So clearly I get $\left( \begin{array}{ccc} 1 & a \\ 0 & 1 \end{array} \right)$ as my answer. However, WolframAlpha says differently:

Wolfram

What's going on?

Answer 1

Most computer programs interpret e^matrix componentwise. In matlab, for example, to do the matrix exponential, you use exp*m*, not exp (which does componentwise exponentiation).

This link tells you how to use MatrixExp in Mathematica (and Wolfram Alpha) to do matrix exponentiation.