# 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.

Tags :