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

Jazmyn Saunders 2022-09-11 Answered
Find 5 × 5 invertible matrix A over F 3 such that A 1 = 2 A 3 + 2 I, A I
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Answers (1)

Marie Horn
Answered 2022-09-12 Author has 12 answers
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.

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