Mathematica

Mathematica is a widely-used computer algebra system originally developed by Stephen Wolfram and sold by his company Wolfram Research. Wolfram started to work on the program in 1986 and released the first version in 1988. The current version is 5.0.

It is available on a wide variety of systems, supports a huge range of operations, and features extensive layout and graphical capabilities.

The Mathematica system is comprised of a kernel, which does the computation with limited user intended output, that can work over a network, on a more powerful computer, for instance, whilst the frontend which comprises of a prettyprinter with a notebook metaphor, can be on any computer and use this external kernel if it so desired.

The frontend system processes input from a user in Mathematica code (see below for an example) and returns the output together with the input each in a cell (as does Maple). Having cells in this way also allows for outlining and sectioning of a document, referred to as a Mathematica notebook. Simple symbolic output can be prettyprinted or be presented back in a similar form to Mathematica code. More complex output such as images and graphs are presented as images in these cells.

Alternatives are the Maple (commercial) and Maxima (free, GPL).

Example Mathematica code

The following Mathematica sequence will find the determinant of the 6×6 matrix whose i, j'th entry contains ij with all zero entries replaced as 1.
 In[1]:= Det[ReplaceAll[Table[i j, {i,0,5}, {j,0,5}],{0->1}]]
 Out[1]:= 0

So the determinant of such a matrix is 0.

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