Find $5\times5$ invertible matrix $A$ over $\mathbb{F}_3$ such that $A^{-1} = 2A^3 +2I$, $A \neq I$

Question

I have tried to solve the above using the cayley hamilton theorem which yields nothing as I get $-1$ which is not in my field.

I feel like I need to do a sub-block decomposition.

Answer 1

The equation can be rewritten (using $2=-1$) to \begin{align} A^4+A+1=0 \end{align} So we can simply choose the Companion Matrix of that polynomial: \begin{align} A = \begin{pmatrix} 0&&&-1\\ 1&0&&0\\ &1&0&0\\ &&1&-1 \end{pmatrix} \end{align},

whoose characteristic polynomial (and also its minimal polynomial) is exactly $x^4+x+1$. Therefore it satisfies the formula.

But that matrix is only $4\times 4$, so you need to add an additional row/column, for example:

\begin{align} A = \begin{pmatrix} 0&&&-1&\\ 1&0&&0&\\ &1&0&0&\\ &&1&-1& \\ &&&&1 \end{pmatrix} \end{align},

which does not change the minimal polynomial, so the equation is still satisfied.