Take, for example, $A$ of order $2 \times 2$. The characteristic equation comes to be $$A^2 - (\text{tr}(A)) A + \det (A) = 0 $$ But is this the only quadratic equation that this matrix satisfies? Can it satisfy infinitely many cubic and biquadratic equation?
Are the results the same for $3 \times 3$ matrices? That is, it satisfies only one cubic equation that is its characteristic equation, satisfy infinite biquadratic equations and may or may not satisfy even one quadratic equation?
What can be said about general $n \times n$ matrices?
Any help (even partial) and even links are appreciated. Thank you for the help. I cannot show my work because I have no idea of how to solve this, sorry.
I have proved that the only quadratic a $2 \times 2$ is its characteristic equation by assuming a general $2\times 2$ matrix and assuming the equation to $A^2 - XA + Y (I) = 0$ , then solving for $X$ and $Y$.Assume that A is not a null matrix.