Show that this matrix is singular

Question

If $\det P=-1$ and $P$ is an orthogonal matrix. Show that $P+I_n$ is singular matrix.

Please help it with only matrix algebra.

Answer 1

Let $\lambda_1,\dots,\lambda_n$ the eigenvalues. Because $P$ is an orthogonal matrix, then $\lambda_j=\pm 1$ for all real eigenvalues. Note that if $\lambda\in\mathbb{C}$ is a complex eigenvalue, then $\overline{\lambda}$ too and $\lambda\overline{\lambda}=\mid \lambda\mid^2=1$

Also, we have $\prod\limits_{j=1}^n\lambda_j=\det P =-1$ and exist $j_0$ such that $\lambda_{j_0}=-1$.

Now, if $J$ is then Jordan form of $P$, then $\det(P+I_n)=\det(J+I_n)=0$ because $J$ has a $-1$ in the diagonal and is inf triang.

Answer 2

Using the fact that $P^TP=I$ and $\det(P)=-1$, we obtain $$ \det(I+P)=\det(P^TP+P)=\det(P)\det(P^T+I)=-\det((I+P)^T)=-\det(I+P)$$ which implies that $\det(I+P)=0$ and hence that $I+P$ is singular.