linear algebra matrices and matrix operations

Question

Show that if a square matrix $A$ satisfies $A^3 + 4A^2 -2A +7I = O$, then so does $A^T$.

Answer 1

$$A^3+4A^2-2A+7I=0\implies 0=0^T=\left(A^3+4A^2-2A+7I\right)^T=$$

$$=(A^3)^T+(4A^2)^T-(2A)^T+(7I)^T$$

and now use that $\;(A^n)^T=(A^T)^n\;$