 Characteristic polynomials of AB and BA (source code)

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= Characteristic polynomials of AB and BA
{c}
{title2=$\chi_{AB}=\chi_{BA}$}

For square matrices $A,B$ over a field,
$$
\det(tI-AB)=\det(tI-BA).
$$
If $A$ is invertible, the products are <similar matrices>. In general, apply that case to $(A-sI)B$ and $B(A-sI)$ for all but finitely many $s$, then use polynomial identity in $s$.

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