Suppose $A$ is a matrix with distinct and positive $n$ eigenvalues. How many real matrices $B$ exist such that $B^k=A$? Suppose $\mathbf{A}$ is an $n\times n$ matrix with distinct and positive $n$ eigenvalues.
How many real matrices $\mathbf{B}$ exist such that $\mathbf{B}^k=\mathbf{A}$?

Maybe diagonalization or Jordan decomposition is useful?

Hint: If $B$ is in Jordan form and $B^k$ has $n$ distinct real eigenvalues, then what can we say about $B$?  Note that $B^k$ will be upper triangular (but not necessarily in Jordan form).
