Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Digital systems, linear algebra, introduction to formal proof (resit) / 5 / d

For $1=(1,1,1{)}^T$, the entries of $P^{T}1$ are the column sums of $P$. Henceso $1$ is an eigenvalue of $P^T$, with eigenvector $1$.

The determinant is unchanged by matrix transposition, soThus $P-I$ has a nontrivial null space, givingThe same argument with $\lambda I$ shows that a matrix and its transpose have the same characteristic polynomial. It does not say that they have the same eigenvectors: here $P1=(0.9,1.2,0.9{)}^T\ne 1$.