If $A$ is an $n\times n$ integer matrix, then trivially $S=A+A^t$ and $P = AA^t$ where $t$ is ``transpose", are both symmetric.

Assume that $A$ is also a "$\lbrace -1,1 \rbrace$" matrix, i.e., the square of each entry in $A$ is equal to $1$.

Is there some rational-coefficient symmetric polynomial $P(x,y)$ (depending possibly on $A$ ?) such that $$ P(A,A^t) $$ is also a $\lbrace -1,1 \rbrace$ matrix?

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