Supertropical matrix algebra
  The objective of this paper is to develop a general algebraic theory of supertropical matrix algebra, extending [11]. Our main results are as follows:   * The tropical determinant (i.e., permanent) is multiplicative when all the determinants involved are tangible.   * There exists an adjoint matrix $\adj{A}$ such that the matrix $A \adj{A}$ behaves much like the identity matrix (times $|A|$).   * Every matrix $A$ is a supertropical root of its Hamilton-Cayley polynomial $f_A$. If these roots are distinct, then $A$ is conjugate (in a certain supertropical sense) to a diagonal matrix.   * The tropical determinant of a matrix $A$ is a ghost iff the rows of $A$ are tropically dependent, iff the columns of $A$ are tropically dependent.   * Every root of $f_A$ is a "supertropical" eigenvalue of $A$ (appropriately defined), and has a tangible supertropical eigenvector. 
Zur Izhakian, Louis Rowen