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. 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.
