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


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.
