Computing Determinant of a Matrix $\textrm{det}(A^{101} - A)$

by Zach   Last Updated November 18, 2018 11:20 AM

Let $A$ be a $n$-by-$n$ matrix with real entries such that $A^{-1} = 3A$. What is the determinant of $A^{101} - A$ i.e. $\textrm {det}(A^{101} - A)$?

My attempt:
$$A^{-1} = 3A \Rightarrow 3A^{2} = I \Rightarrow A^{2} = \frac{1}{3}I$$ \begin{align} \textrm{det}(A^{101} - A) & = \textrm{det}[A(A^{100}-I)] \\ & = \textrm{det}(A)\textrm{det}(A^{100}-I) \\ & = \textrm{det}(A)\textrm{det}[(A^{2})^{50}-I] \\ & = \textrm{det}(A)\textrm{det}[(\frac{1}{3}I)^{50}-I] \\ & = \textrm{det}(A)\textrm{det}[(\frac{1}{3^{50}})I-I] \\ & = \frac{1}{3^{n}}\textrm{det}(A^{-1})\textrm{det}[(\frac{1}{3^{50}})I-I] \\ \end{align} Any hints on how to continue from here will be appreciated.



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