Show that if for a square matrix $A$ satisfies $A^3+ 4A^2 - 2A + 7I = 0 $ then so does $A^T$
I suspect that this is not always true so I need to do an example not the general case ( I partially suspect this cause the next question asks me if this is always true or sometimes false so I need to find a counter-example as well)
not really sure where to start this problem.
ah okay, in my notes I have $(A^{-1})^T = (A^T)^{-1}$ but I didn't know it worked for powers, not just inverses. So then it is always true that if a matrix A satisfies that equation then $A^T$ will as well?
Does A have to be symmetric for this to work?