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Find all matrix in $\mathbb{F}_{13}$

Source: Brazil Undergrad MO 2021

March 15, 2021

linear algebramatrixBrazilian Undergrad MOBrazilian Undergrad MO 2020number theory

Problem Statement

Let

\mathbb{F}_{13} = {\overline{0}, \overline{1}, \cdots, \overline{12}}

be the finite field with

13

elements (with sum and product modulus

13

). Find how many matrix

A

of size

5

5

with entries in

\mathbb{F}_{13}

exist such that

A^5 = I

where

I

is the identity matrix of order

5