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Let $F = \mathbb{F}_3$ and let $n$ be a positive integer. Let $D = [d_{ij} ]\in\mathscr{M}_{n×n}(F)$ be a nonsingular diagonal matrix and let $A\in\mathscr{M}_{n×n}(F)$. Show that $1\notin\operatorname{spec}(DA)$ if and only if $D - A$ is nonsingular. (Exercise 674 from Golan, The Linear Algebra a Beginning Graduate Student Ought to Know.)

$(\Rightarrow)$

I found that $I-DA$ is nonsingular but then I gent that $D^{-1}-A$ is nonsingular. I don't know how to use the fact that $D$ is diagonal to justify that $D-A$ is nonsingular.

I also know that $1\notin\operatorname{spec}((DA)^{-1})$

Thanks

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  • $\begingroup$ $D$ may have to be a scalar multiple of identity... $\endgroup$
    – DVD
    May 23 '15 at 22:39
  • $\begingroup$ what is the reason of that? $\endgroup$
    – José
    May 23 '15 at 22:43
  • $\begingroup$ For $A$ and $D$ to commute... $\endgroup$
    – DVD
    May 23 '15 at 22:50
  • $\begingroup$ I don't see why they commute $\endgroup$
    – José
    May 23 '15 at 22:55
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Hint: $D^{-1}=D$ since $F=\mathbb{F}_3$ and $D$ is diagonal.

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  • $\begingroup$ I take it you solved the problem in the mean time. $\endgroup$
    – Git Gud
    May 23 '15 at 22:50
  • $\begingroup$ I just realized of that. $\endgroup$
    – José
    May 23 '15 at 22:52

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