If $f(x)$ = det($xI-A$), what is the determinant of the matrix $aA-bI$. I need to express my answer in terms of $f(x), a$ and $b$.
2026-04-09 06:11:40.1775715100
Characteristic polynomial of $aA$-$bI$, $a,b \in$ $R$
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$$\begin{align}\det(aA-bI) &= \det\left(a\left(A-\frac ba I\right)\right) \\ &= (-a)^n\det\left(\frac ba I-A\right) \\ &= (-a)^n f\left(\frac ba\right)\end{align}$$