  The only invertible matrices in tropical algebra are diagonal matrices,
permutation matrices and their products. However, the pseudo-inverse
$A^\nabla$, defined as $\frac{adj(A)}{det(A)}$, with $det(A)$ being the
tropical permanent (also called the tropical determinant) of a matrix $A$,
inherits some classical algebraic properties and has some surprising new ones.
Defining $B$ and $B'$ to be tropically similar if $B' =A^\nabla BA$, we examine
the characteristic (max-)polynomials of tropically similar matrices as well as
those of pseudo-inverses. Other miscellaneous results include a new proof of
the identity for $det(AB)$ and a connection to stabilization of the powers of
definite matrices.
