A matrix inverse can be found by creating a system of linear equations from the matrix and solving for the unknowns. There are two classical methods that create a system of n^2 equations and unknowns from an nxn matrix. The question is which choices of n^2 equations from the 2n^2 total equations yield the unique solution for the inverse. Sage is used to perform Gaussian elimination on different choices of systems of equations for various matrix sizes to determine which choices provide a valid solution. Possible applications include cryptography, where data could be encoded in a matrix or its inverse and "encrypted" by not revealing the choice of equations used.

1. Denzel Akosa
Professor Lauve
Mathematics
08-07-14
A matrix is a function whose inverse (when it exists) can be obtained by creating a system of
equations and solving for them using unknowns. Two classical methods of finding the inverse
creates a system of linear equations (in each, an nxn matrix corresponds to a system of n^2
equations and unknowns). We ask the question first posed by famous mathematician and
computer scientist Schutzenberger over twenty years ago: from the 2n^2 equations, which
choices of n^2 equations yield the (unique) solution for the inverse function? As the size of the
matrix increases from 2x2 to 4x4 and so on, solving for the matrix inverse by hand becomes
intractable. We use the computer programming language Sage to perform Gaussian elimination
on the various choices of systems of equations, for various matrix sizes. The secondary goal of
this project is to look for patterns explaining why any given choice of equations yields a valid
solution or not.
Possible applications are to cryptography. E.g., data may be encoded in the matrix or its inverse,
and it is “encrypted” because the eavesdropper does not know which choice of n^2 equations
was passed from the sender to the receiver.