Prove (or find a counterexample) that a regular probability matrix, aka a left stochastic matrix, has only one linearly independent eigenvector where $\lambda = 1$. In other words, prove that the geometric multiplicity of $\lambda=1$ is one.
The standard definition of a probability matrix is a square matrix with all real, non-negative values, where all columns sum to $1$. A regular probability matrix has all strictly positive, non-zero values.
I see a simple proof that $1$ is an eigenvalue of $A$. With non-regular probability matrices like the identity matrix which have zero values, there clearly are multiple eigenvectors with eigenvalue $1$.