How to prove that $P+ I_n$ is a singular matrix?

Question

$P$ is an $n \times n$ real orthogonal matrix with $\det P= -1$, Prove that $P + I_n$ is a singular matrix.

Answer 1

Eigenvalues of an orthogonal matrix have modulus $1$.

Given that $\det P =-1$, we know that one of the eigenvalues is $-1$.

Let the corresponding eigenvector be $v$.

$$(P+I_n)v=Pv+v=-v+v=0$$

Hence $P+I_n$ is singular.

Answer 2

Eigenvalues of an orthogonal matrix have modulus 1. But they are either real (i.e. $\pm 1$) or come in complex pairs. The product of a complex pair is +1 so there must be at least one eigenvalue -1 or else the product of eigenvalues is +1. And then $P+I$ has an eigenvalue zero.