## Abstract

Let F be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices (Formula Presented) and for the whole matrix space M _{n}(F). It is known that for n = 2, there are bijective linear maps Φ on (Formula Presented) and M _{n}(F) satisfying the condition per A = det Φ(A). As an application of the obtained results, we show that if n ≥ 3, then the situation is completely different and already for n = 3, there is no pair of maps (Φ, φ{symbol}), where Φ is an arbitrary bijective map on matrices and (Formula Presented) is an arbitrary map such that per A = φ{symbol}(det Φ(A)) for all matrices A from the spaces(Formula Presented) and M _{n}(F), respectively. Moreover, for the space M _{n}(F), we show that such a pair of transformations does not exist also for an arbitrary n > 3 if the field F contains sufficiently many elements (depending on n). Our results are illustrated by a number of examples.

Original language | English |
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Pages (from-to) | 404-413 |

Number of pages | 10 |

Journal | Journal of Mathematical Sciences |

Volume | 193 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2013 |

Externally published | Yes |

### Bibliographical note

Funding Information:Acknowledgements. This research was partially supported by a joint Slovene–Russian grant BI-RU/08-09-009. The research of the second author is also supported by the Russian Foundation for Basic Research (project No. 09-01-00303a).