With the multiplicative map theory on matrix and its applications, a lot of attempts have been made for judging whether a multiplicative map can preserve certain desired numerical characters and obtain the explicit form of a multiplicative map under the restriction of perserving some numerical characters. In this respect, multiplicative maps without assuming linearity on matrix algebra, which have certain rank preserving or norm preserving properties, are considered mainly in this paper. By virtue of a way of construction, the complete descriptions of those maps are presented, and it is shown that a maximum column sum norm preserving multiplicative map is one of Frobenius norm and a Frobenius norm preserving multiplicative map must preserve spectral radius, numerical radius, normality, unitarity etc.. In particular, a new approach is also provided for judging whether a multiplicative map preserves rank.