  Can the cross product be generalized? Why are the trace and determinant so
important in matrix theory? What do all the coefficients of the characteristic
polynomial represent? This paper describes a technique for `doodling' equations
from linear algebra that offers elegant solutions to all these questions. The
doodles, known as trace diagrams, are graphs labeled by matrices which have a
correspondence to multilinear functions. This correspondence permits
computations in linear algebra to be performed using diagrams. The result is an
elegant theory from which standard constructions of linear algebra such as the
determinant, the trace, the adjugate matrix, Cramer's rule, and the
characteristic polynomial arise naturally. Using the diagrams, it is easy to
see how little structure gives rise to these various results, as they all can
be `traced' back to the definition of the determinant and inner product.
