From another thread:

If $p$ is a polynomial without real roots, and $A$ is a real matrix with odd dimension, then $p(A) = 0 $ never holds.

This follows mainly from the fact that odd degree peal polynomials always have a root.

From another thread:

If $p$ is a polynomial without real roots, and $A$ is a real matrix with odd dimension, then $p(A) = 0 $ never holds.

This follows mainly from the fact that odd degree real polynomials always have a root.