Cramer and Cayley-Hamilton in the max algebra (Q1112131)

The authors consider a so-called max algebra, i.e. a semiring of the real numbers with added -\(\infty\) with two binary operations: \(+\) is the maximum operation with respect to the usual ordering, * is the usual addition. It is shown that Cramer's rule and the Cayley-Hamilton theorem hold true in the max algebra. A role of the determinant in the classical case is replaced by the permanent. The equation \(A*x=b\) with A as a nonsingular square matrix not always has solutions. Solutions given by this new Cramer's rule are generally non-unique. This interesting paper is illustrated by many examples.