Minimal and characteristic polynomial of matrix transpose operator

[specialnlovin]

Let V =M_n(k),n>1 and T:V→V defined by T(M)=M^t (transpose of M).
i) Find the minimal polynomial of T. Is T diagonalisable when k = R,C,F₂?
ii) Suppose k = R. Find the characteristic polynomial ch_T .
I know that T²=T(M^t))=M and that has got to help me find the minimal polynomial

 

[micromass]

You'll need to find a polynomial $$p(x)=x^n+...+a_1x+a_0$$, such that p(T)=0, i.e.

$$T^n+...+a_1T+a_0I=0$$

You know that $$T^2(M)=M$$, can you use this to find a suitable polynomial??

 

[specialnlovin]

All i can think of is f(x)=1 or f(T)=I

 

[micromass]

Come on, how do you write $$T^2(M)=M$$ as a polynomial in T??