Powers, inverse, eigenvalues and diagonalization of $\begin{pmatrix}1&2\\5&4\end{pmatrix}$

Consider the matrix \(A = { \left[ \begin{array} { l l } { 1 } & { 2 } \\ { 5 } & { 4 } \end{array} \right] }\)

(a)Find \(A ^ { T } , A ^ { 2 } , A ^ { 3 } , A ^ { - 1 }\) , TrA, det A.

2. Find the eigenvalues and the eigenvectors of A.

3. Write the transformation that reduces A to diagonal form.