Show that minimal polynomial for $n\times n$ matrix and its transpose is the same

Question

Show that minimal polynomial for $n\times n$ matrix and its transpose is the same

Answer 1

It is essentially that simple. You've shown that $u(A^t) = 0$ so the minimal polynomial of $A^t$ must divide $u$. The same proof, starting with $A^t$ instead of $A$, shows that the minimal polynomial of $A$ divides the minimal polynomial of $A^t$. Finally if two monic polynomials divide each other, then they must be equal.