The question is as follows:
Define $T : M_n(\mathbb{R}) → M_n(\mathbb{R})$ by $T(A) = A^t$. Prove that T has two distinct eigenvalues, and that its eigenvectors span $M_n(\mathbb{R})$.
I am completely stuck on this. I have an intuitive idea of why it is correct but no proof. I have tried explaining how 1 and -1 are the only values that make the determinant of $A-tI$ equal to $0$ by using ideas about addition and swapping of columns using EROs. I've tried to show it is diagonalizable, and therefore has enough linearly independent vectors to span $M_n(\mathbb{R})$, all to no avail. If somebody could help me with this I'd be extremely thankful.
Hint:
What is minimal polynomial of the transpose map in $M_n(\mathbf R)$? Which criteria for diagonalisability do you know?