For $n > 1$, the maximum multiplicity of any eigenvalue of an $n \times n$ matrix with elements from $\mathbb{R}$ is

For an $n \times n$ matrix $A$, the characteristic polynomial $\det(A - \lambda I) = 0$ is of degree $n$.

By the Fundamental Theorem of Algebra, a polynomial of degree $n$ has at most $n$ roots (counting multiplicities). Therefore, the algebraic multiplicity of any single eigenvalue can be at most $n$.

For example, the $n \times n$ identity matrix $I_n$ has a single eigenvalue $\lambda = 1$ with algebraic and geometric multiplicity equal to $n$.

Hence, the maximum multiplicity is $n$, which corresponds to Option (A).