is it true that $\det(I+A)>0$ , if $\det(A)>0$?

Question

I saw an inequality for $n\times n$ matrices. I was wondering if the inequality is true or not?

Does $\det(A)>0$ imply $\det(I+A)>0$?

Asaf Karagila · Accepted Answer · 2013-09-28 14:37:35Z

11

Consider $-I$ in $M_n(\Bbb R)$ for an even $n$.

answered Sep 28, 2013 at 14:37

Asaf Karagila♦

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$\begingroup$ @Javier: it depends on whether you think $\begin{bmatrix} -1 & \,0 \\ \,0 & -1 \end{bmatrix}$ is simpler or the same $\endgroup$
– Henry
Sep 28, 2013 at 19:10
$\begingroup$ @Henry: Well, for $n=2$ this this $-I$... $\endgroup$
– Asaf Karagila ♦
Sep 28, 2013 at 22:25

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mrf · Accepted Answer · 2013-09-28 14:33:08Z

8

Let $$ A = \begin{bmatrix} -3 & 0 \\ 0 & -1/2 \end{bmatrix}.$$

answered Sep 28, 2013 at 14:33

mrf

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2 Answers 2