Find 5 × 5 invertible matrix A over F 3 such that A − 1 = 2 A 3 + 2 I, A ≠ I

Question

Find   5  ×  5 invertible matrix A over             F        3   such that       A          −      1        =  2      A    3    +  2  I,   A  ≠  I

Marie Horn · Accepted Answer

The equation can be rewritten (using 2=−1) to                              A          4                +        A        +        1        =        0            So we can simply choose the Companion Matrix of that polynomial:                    A        =                  (                                                    0                                                                        −                1                                                                    1                                            0                                                          0                                                                                  1                                            0                                            0                                                                                                1                                            −                1                                              )                    whoose characteristic polynomial (and also its minimal polynomial) is exactly       x    4    +  x  +  1. Therefore it satisfies the formula.But that matrix is only   4  ×  4, so you need to add an additional row/column, for example:                    A        =                  (                                                    0                                                                        −                1                                                                                  1                                            0                                                          0                                                                                                1                                            0                                            0                                                                                                              1                                            −                1                                                                                                                                          1                                              )                    which does not change the minimal polynomial, so the equation is still satisfied.

Find 5 xx 5 invertible matrix A over mathbbF_3 such that A^(-1)=2A^3+2I, A != I

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