# 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 ${\mathbb{F}}_{3}$ such that ${A}^{-1}=2{A}^{3}+2I$, $A\ne I$
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Marie Horn
The equation can be rewritten (using 2=−1) to
$\begin{array}{r}{A}^{4}+A+1=0\end{array}$
So we can simply choose the Companion Matrix of that polynomial:
$\begin{array}{r}A=\left(\begin{array}{cccc}0& & & -1\\ 1& 0& & 0\\ & 1& 0& 0\\ & & 1& -1\end{array}\right)\end{array}$
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:
$\begin{array}{r}A=\left(\begin{array}{ccccc}0& & & -1& \\ 1& 0& & 0& \\ & 1& 0& 0& \\ & & 1& -1& \\ & & & & 1\end{array}\right)\end{array}$
which does not change the minimal polynomial, so the equation is still satisfied.

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