Let $D$ be a nonsingular diagonal matrix. Show that $1\notin spec(DA)$ if and only if $D - A$ is nonsingular.

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

• $D$ may have to be a scalar multiple of identity...
– DVD
May 23 '15 at 22:39
• what is the reason of that?
– José
May 23 '15 at 22:43
• For $A$ and $D$ to commute...
– DVD
May 23 '15 at 22:50
• I don't see why they commute
– José
May 23 '15 at 22:55

Hint: $D^{-1}=D$ since $F=\mathbb{F}_3$ and $D$ is diagonal.