Given square real matrix $A$ with $\det(A) = 108$ and $(A-2I)(A^2-9I)=0$, is $A$ normal?

Question

Given real square matrix $A$ with $\det (A) = 108$ and $(A-2I)(A^2-9I) = 0$, find:

a. The characteristic and minimal polynomial options (all options).

b. Is $A$ normal?

I think I found the minimal and characteristic polynomials but I can't tell if $A$ is normal.

Answer 1

Hint: Normal matrix with real eigenvalues is Hermitian which in real case just means symmetric. From the minimal polynomial you know how Jordan normal form looks like. Can you find non-symmetric matrix with such a Jordan normal form?